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MIXING OF SPECIES IN A TWO-STREAM SHEAR LAYER
FORCED BY AN OSCILLATING AIRFOIL

presented by

GREGORY JAMES KATCH

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MIXING OF SPECIES IN A TWO-STREAM SHEAR LAYER
FORCED BY AN OSCILLATING AIRFOIL
By

Gregory James Katch

A THESIS

Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of

MASTER OF SCIENCE

Department of Mechanical Engineering

1994

ABSTRACT

MIXING OF SPECIES IN A TWO-STREAM SHEAR LAYER
FORCED BY AN OSCILLATING AIRFOIL

By

Gregory James Katch

The modifications to the structure and the mixing field in a shear layer forced by
an oscillating airfoil were documented over a range of frequencies, amplitudes and
streamwise locations using laser induced fluorescence in the passive scalar mode. The
shear layer structure and growth were observed to conform to the well-established
behavior of forced shear layers. Results on the mixing field show a variation in the
amount of mixed fluid in the central portion of the forced shear layer with increasing
downstream distance and forcing frequency, an increase in the width of the region
occupied by mixed fluid and an increase in the total amount of mixed fluid in the layer.
Only for the highest forcing frequency examined in this study was there an increase in
the fraction of the layer occupied by mixed fluid when compared to the unforced layer.
These results are consistent with those from a shear layer forced by a different method
(oscillating one freestream). Attempts were made to describe the downstream evolution
of the forced mixing layer in terms of a universal non-dimensional curve. The
predominate mixed fluid concentration in the forced shear layer Showed similar behavior

to the layer forced by oscillating one freestream.

Copyright by
Gregory James Katch
1994

to my parents James and Barbara

iv

 

ACKNOWLEDGMENTS

I am grateful to my advisor, Dr. Manooch Koochesfahani, for the opportunity to
learn from his example as an outstanding scientist and educator. Additional thanks goes
to C. G. MacKinnon for all of his instruction and patience. I would also like to thank Dr.
J. F. Foss and Dr. M. Zhuang for their review of this work. Finally, I would like to thank
my family for their support and encouragement throughout the pursuit of this degree.

This work was supported by the Air Force Office of Scientific Research Grant No.

AFOSR-87-0332 and AFOSR-91-O314.

 

TABLE OF CONTENTS

LIST OF TABLES ............................................. viii
LIST OF FIGURES ............................................. ix
LIST OF SYlVIBOLS .......................................... xvii
Chapter 1. INTRODUCTION .................................... 1
1.1 The natural shear layer ................................... 2

1.2 The forced shear layer ................................... 3

1.3 Mixing in the shear layer ................................. 5

1.4 Objectives ............................................ 8
Chapter 2. EXPERIMENTAL FACILITY AND INSTRUMENTATION ..... 10
2.1 Experimental facility ................................... 10

2.2 Forcing mechanism .................................... 12

2.3 Diagnostics .......................................... 13

2.4 Data reduction ........................................ 17
Chapter 3. RESULTS AND DISCUSSION ......................... 22
3.1 Flow visualization results ................................ 22

3.2 Transverse profiles of E, pm, [)0 and p1 ....................... 29

3.2.1 Streamwise dependence ............................. 30

vi

 

3.2.2 Forcing amplitude dependence ........................ 42

3.3 Streamwise variation of 51, 5m, 8m/81 and (pm)max ............. 46

3.4 Nondirnensional streamwise variation of 81, 8m, 8m/81 and (pm)max . . 55

3.5 Probability density function of 2‘, _ ........................... 62

3.5.1 Streamwise variation of the total pdf .................... 63

3.5.2 Amplitude dependence of the total pdf ................... 66

3.6 Comparison with Koochesfahani and MacKinnon (1991) ........... 68

3.7 Comparison with Roberts and Roshko (1985) .................. 72
Chapter 4. CONCLUSIONS .................................... 76
Appendix A. Amplitude dependence of the transverse profiles .............. 78
Appendix B. Distribution of mixed fluid composition .................... 94
Appendix C. Streamwise evolution of the total pdf ..................... 105
Appendix D. Amplitude dependence of the total pdf .................... 109
REFERENCES .............................................. 118

vii

 

LIST OF TABLES

Table 1. Forcing frequency-amplitude combinations. ................... 13

Table 2. Summary of shear layer width, mixed fluid thicknesses and mixed fluid
fraction for the natural and unforced layers. ................... 69

Table 3. Summary of shear layer width, mixed fluid thickness and mixed fluid
fraction for the different forcing conditions. ................... 71

viii

Figure 1.

Figure 2.

Figure 3.

Figure 4.

Figure 5.
Figure 6.
Figure 7.

Figure 8.

Figure 9.

Figure 10.

Figure l 1.

Figure 12.

Figure 13.

LIST OF FIGURES

Schematic of the plane shear layer. ........................

Effect of forcing on the shear layer growth, reprinted from
Browand and Ho (1983). ...............................

The mixing transition, reprinted from Roshko (1990). ...........

Effect of the velocity ration on the mixing transition, reprinted
from Breidenthal (1978). ...............................

Schematic of the shear layer facility. ......................
Schematic of the forcing arrangement. .....................
Schematic of the optical arrangement. .....................

Time series of the unforced shear layer, (a) and (b) are not
consecutive in time, flow is from right to left, At = 1/30 sec .....

Effect of the forcing amplitude on the structure of the shear layer
forced at f = 2 Hz, flow is from right to left, At = 1/30 sec ......

Effect of the forcing amplitude on the structure of the shear layer
forced at f = 4 Hz, flow is from right to left, At = 1/30 sec ......

Effect of the forcing amplitude on the structure of the shear layer
forced at f = 8 Hz, flow is from right to left, At = 1/30 sec ......

Effect of the forcing frequency on the structure of the shear layer
forced at A = 4°, flow is from right to left, At = 1/30 sec. .......

Transverse profiles for the unforced shear layer as a function of the
downstream distance x ................................

ix

2

4

6

ll

12

15

25

26

27

28

31

Figure 14.

Figure 15.

Figure 16.

Figure 17.

Figure 18.

Figure 19.

Figure 20.

Figure 21.

Figure 21.

Figure 23.

Figure 24.

Figure 25.

Figure 26.

Figure 27.

Transverse profiles for the shear layer forced at f = 2 Hz, A = 2°
as a function of the nondimensional downstream distance x" ...... 32

Transverse profiles for the shear layer forced at f = 2 Hz, A = 4°
as a function of the nondimensional downstream distance x’" ...... 33

Transverse profiles for the shear layer forced at f = 2 Hz, A = 8°
as a function of the nondimensional downstream distance x* ...... 34

Transverse profiles for the shear layer forced at f = 4 Hz, A = 2°
as a function of the nondimensional downstream distance x* ...... 35

Transverse profiles for the shear layer forced at f = 4 Hz, A = 4°
as a function of the nondimensional downstream distance x" ...... 36

Transverse profiles for the shear layer forced at f = 4 Hz, A = 6°
as a function of the nondimensional downstream distance x* ...... 37

Transverse profiles for the shear layer forced at f = 8 Hz, A = 2°
as a function of the nondimensional downstream distance x" ...... 38

Transverse profiles for the shear layer forced at f = 8 Hz, A = 4°
as a function of the nondimensional downstream distance x’ ...... 39

Transverse profiles for the shear layer forced at f = 8 Hz, A = 6°
as a function of the nondimensional downstream distance x" ...... 40

Transverse profiles for the shear layer forced at f = 2 Hz as a
function of amplitude A, at x’“ = 0.44 (x = 20.0 cm), compared to
the unforced case ................................... 43

Transverse profiles for the shear layer forced at f = 4 Hz as a
function of amplitude A, at x” = 0.88 (x = 20.0 cm), compared to
the unforced case ................................... 44

Transverse profiles for the shear layer forced at f = 8 Hz as a
function of amplitude A, at x" = 1.78 (x = 20.0 cm), compared to
the unforced case ................................... 45

Streamwise evolution of the shear layer width 5, as a function of
the forcing amplitude A compared to the unforced case ......... 47

Streamwise evolution of the mixed fluid thickness 5", as a function
of the forcing amplitude A compared to the unforced case ....... 49

Figure 28.

Figure 29.

Figure 30.

Figure 31.

Figure 32.

Figure 33.

Figure 34.

Figure 35.

Figure 36.

Figure 37.

Figure 38.

Figure 39.

Figure 40.

Figure 42.

Streamwise evolution of the mixed fluid fraction fin/51 as a
function of the forcing amplitude A compared to the unforced
case ............................................. 51

Streamwise evolution of the amount of mixed fluid in the center of
the layer as a function of the forcing amplitude A compared to the
unforced case ...................................... 53

Nondirnensional streamwise evolution of the shear layer width
normalized by the forcing wavelength for the various forcing
conditions ......................................... 55

Nondirnensional streamwise evolution of the mixed fluid thickness
normalized by the forcing wavelength for the various forcing

conditions ......................................... 56

Nondimensional streamwise evolution of the mixed fluid fraction
for the various forcing conditions ........................ 57

Nondirnensional streamwise evolution of the amount of mixed fluid
in the center of the layer for the various forcing conditions ...... 58

Streamwise evolution of the concentration thickness as a function
of the forcing amplitude A compared to the unforced case ....... 59

Nondirnensional streamwise evolution of the concentration
thickness normalized by the forcing wavelength for the various
forcing conditions ................................... 61
Nondirnensional streamwise evolution of the mixed fluid thickness
normalized by the concentration thickness for the various forcing
conditions ......................................... 62

Streamwise evolution of the total pdf of the unforced shear layer . . 64

Nondirnensional streamwise evolution of the total pdf for the shear
layer forced at f = 2 Hz, A = 4° ......................... 65

Nondirnensional streamwise evolution of the total pdf for the shear
layer forced at f = 4 Hz, A = 4° ......................... 65

Nondimensional streamwise evolution of the total pdf for the shear
layer forced at f = 8 Hz, A = 4° .......... _ ............... 66

Amplitude dependence of the total pdf for the shear layer forced at

xi

Figure 43.

Figure 44.

Figure 45.

Figure 46.

Figure 47.

Figure A.l.

Figure A.2.

Figure A.3.

Figure A.4.

Figure A.5.

Figure A.6.

f = 4 Hz, at x = 20.0 cm (x* = 0.88), compared to the
unforced case ......................................

Amplitude dependence of the total pdf for the shear layer forced at
f = 8 Hz, at x = 20.0 cm (x* = 1.78), compared to the unforced
case .............................................

Nondirnensional streamwise (frequency) dependence of the mixed
fluid fraction for various forced cases compared with
Koochesfahani and M°°Kinnon (1991) .....................

Ratio of forced product thickness to natural product thickness as a
function of x’" : low Reynolds number case, reprinted from Roberts
(1985) ...........................................

Ratio of forced product thickness to natural product thickness as a
function of x' : high Reynolds number case, reprinted from Roberts
(1985) ...........................................

Ratio of forced product thickness to unforced product thickness as
a function of x* .....................................

Transverse profiles for the shear layer forced at f = 2 Hz as a
function of the forcing amplitude A, at x’“ = 0.28, compared to the
unforced case ......................................

Transverse profiles for the shear layer forced at f = 2 Hz as a
function of the forcing amplitude A, at x* = 0.31, compared to the
unforced case ......................................

Transverse profiles for the shear layer forced at f = 2 Hz as a
function of the forcing amplitude A, at x” = 0.34, compared to the
unforced case ......................................

Transverse profiles for the shear layer forced at f = 2 Hz as a
function of the forcing amplitude A, at x* = 0.37, compared to the
unforced case ......................................

Transverse profiles for the shear layer forced at f = 2 Hz as a
function of the forcing amplitude A, at x* = 0.41, compared to the
unforced case ......................................

Transverse profiles for the shear layer forced at f = 4 Hz as a

function of the forcing amplitude A, at x* = 0.56, compared to the
unforced case ......................................

xii

70

74

74

75

Figure A.7.

Figure A.8.

Figure A.9.

Figure A. 10.

Figure Al 1.

Figure A. 12.

Figure A. 13.

Figure A. 14.

Figure A. 15.

Figure B.1.

Figure B.2.

Figure B.3.

Figure B.4.

Transverse profiles for the shear layer forced at f = 4 Hz as a
function of the forcing amplitude A, at x’" = 0.62, compared to the
unforced case ......................................

Transverse profiles for the shear layer forced at f = 4 Hz as a
function of the forcing amplitude A, at x' = 0.68, compared to the
unforced case ......................................

Transverse profiles for the shear layer forced at f = 4 Hz as a
function of the forcing amplitude A, at x’" = 0.75, compared to the
unforced case ......................................

Transverse profiles for the shear layer forced at f = 4 Hz as a
function of the forcing amplitude A, at x’“ = 0.82, compared to the
unforced case ......................................

Transverse profiles for the shear layer forced at f = 8 Hz as a
function of the forcing amplitude A, at x‘ = 1.11, compared to the
unforced case ......................................

Transverse profiles for the shear layer forced at f = 8 Hz as a
function of the forcing amplitude A, at x” = 1.24, compared to the
unforced case ......................................

Transverse profiles for the shear layer forced at f = 8 Hz as a
function of the forcing amplitude A, at x'“ = 1.38, compared to the
unforced case ......................................
Transverse profiles for the shear layer forced at f = 8 Hz as a
function of the forcing amplitude A, at x* = 1.51, compared to the
unforced case ......................................
Transverse profiles for the shear layer forced at f = 8 Hz as a
function of the forcing amplitude A, at x" = 1.64, compared to the
unforced case ......................................

Composition distribution for the unforced shear layer. ..........

Composition distribution for the shear layer forced at f = 2 Hz,
A = 2° ...........................................

Composition distribution for the shear layer forced at f = 2 Hz,
A = 4° ...........................................

Composition distribution for the shear layer forced at f = 2 Hz,

xiii

95

A=8° ............................................ 98

Figure B.5. Composition distribution for the shear layer forced at f = 4 Hz,
A = 2° ........................................... 99

Figure B.6. Composition distribution for the shear layer forced at f = 4 Hz,
A = 4° .......................................... 100

Figure B.7. Composition distribution for the shear layer forced at f = 4 Hz,
A = 6° .......................................... 101

Figure B.8. Composition distribution for the shear layer forced at f = 8 Hz,

Figure B.9. Composition distribution for the shear layer forced at f = 8 Hz,
A = 4° .......................................... 103

Figure B.10. Composition distribution for the shear layer forced at f = 8 Hz,
A = 6° .......................................... 104

Figure C.1. Streamwise evolution of the total pdf for the shear layer forced at
f=2Hz,A=2° ................................... 106

Figure C.2. Streamwise evolution of the total pdf for the shear layer forced at
f=2Hz,A=8° ................................... 106

Figure C.3. Streamwise evolution of the total pdf for the shear layer forced at
f=4Hz,A=2° ................................... 107

Figure C.4. Streamwise evolution of the total pdf for the shear layer forced at
f=4Hz,A=6° ................................... 107

Figure C.5. Streamwise evolution of the total pdf for the shear layer forced at
f=8Hz,A=2° ................................... 108

Figure C.6. Streamwise evolution of the total pdf for the shear layer forced at
f=8Hz,A=6° ................................... 108

Figure D. 1. Amplitude dependence of the total pdf for the shear layer forced at
f = 2 Hz, at x = 12.5 cm (x' = 0.28), compared to the unforced
case ............................................ l 10

Figure D.2. Amplitude dependence of the total pdf for the shear layer forced at

f = 2 Hz, at x = 14.0 cm (x* = 0.31), compared to the unforced
case ............................................ 110

xiv

Figure D.3.

Figure D.4.

Figure D.5.

Figure D.6.

Figure D.7.

Figure D.8.

Figure D.9.

Figure D. 10.

Figure D.11.

Figure D. 12.

Figure D.13.

Figure D.14.

Amplitude dependence of the total pdf for the shear layer forced at
f = 2 Hz, at x = 15.5 cm (x* = 0.34), compared to the unforced
case ............................................

Amplitude dependence of the total pdf for the shear layer forced at
f = 2 Hz, at x = 17.0 cm (x* = 0.37), compared to the unforced
case ............................................

Amplitude dependence of the total pdf for the shear layer forced at
f = 2 Hz, at x = 18.5 cm x” = 0.41), compared to the unforced
case ............................................

Amplitude dependence of the total pdf for the shear layer forced at
f = 4 Hz, at x = 12.5 cm (x* = 0.56), compared to the unforced
case ............................................

Amplitude dependence of the total pdf for the shear layer forced at
f = 4 Hz, at x = 14.0 cm (x* = 0.62), compared to the unforced
case ............................................

Amplitude dependence of the total pdf for the shear layer forced at
f = 4 Hz, at x = 15.5 cm (x"' = 0.68), compared to the unforced
case ............................................

Amplitude dependence of the total pdf for the shear layer forced at
f = 4 Hz, at x = 17.0 cm (x* = 0.75), compared to the unforced
case ............................................

Amplitude dependence of the total pdf for the shear layer forced at
f = 4 Hz, at x = 18.5 cm (x* = 0.82), compared to the unforced
case ............................................

Amplitude dependence of the total pdf for the shear layer forced at
f = 8 Hz, at x = 12.5 cm (x* = 1.11), compared to the unforced
case ............................................

Amplitude dependence of the total pdf for the shear layer forced at
f = 8 Hz, at x = 14.0 cm (x’ = 1.24), compared to the unforced
case ............................................
Amplitude dependence of the total pdf for the shear layer forced at
f = 8 Hz, at x = 15.5 cm (x* = 1.38), compared to the unforced

case ............................................

Amplitude dependence of the total pdf for the shear layer forced at

XV

f = 8 Hz, at x = 17.0 cm (x‘ = 1.51), compared to the
unforced case

Figure D. 15. Amplitude dependence of the total pdf for the shear layer forced at

f = 8 Hz, at x = 18.5 cm (x’ = 1.64), compared to the unforced
case

xvi

 

 

 

Symbol

cal
corr

COIT 0

p(§,y)

LIST OF SYMBOLS

Description

amplitude of oscillation

airfoil span

average product concentration
airfoil chord

instantaneous dye concentration
freestream dye concentration
coefficient of mass diffusivity
forcing frequency

natural shear layer roll up frequency
half height of the test section

raw instantaneous intensity
intensity in the calibration image
corrected instantaneous intensity
corrected freestream dye intensity

probability density function of g at y

xvii

 

pa 0)

p, 0)
pm 0)
(paw
P ( 5 )

,.

Re,Sl
Re ’
Sc
At

AU

Greek symbols

probability of pure low-speed fluid at y
probability of pure high-speed fluid at y
probability of mixed fluid at y

maximum value of the pm (y) curve

average probability density function of l;
velocity ratio, U ,/ U2

Reynolds number, AU 51 / v

unit Reynolds number AU / v

Schmidt number, v/D

time between fields

velocity difference, U 1 - U2

high-speed velocity

low-speed velocity

convection speed, average speed, (U 1 + U2 )/ 2
high-speed fluid volume

low-speed fluid volume

streamwise coordinate

nondimensional streamwise parameter, Ax f / Uc

vertical coordinate

absorption coefficient

shear layer 1% thickness

xviii

 

mixed fluid thickness

shear layer visual thickness
chemical product thickness
uncertainty in E,

dye molar absorption coefficient
momentum thickness
concentration thickness

growth rate parameter, (U, - U2 )/(U, + U2)
diffusion scale

Kolmogorov scale, 81 Re "3’4
kinematic viscosity
concentration

average concentration

xix

Chapter 1

INTRODUCTION

The shear layer has been the subject of a large number of studies over the past
few decades. One reason for the great interest in this flow is the common occurrence of
shear layers in both naturally occurring phenomena and in industrial processes. Some
familiar examples of shear layers are the wind blowing over a body of water, the injection
of fuel into a combustor and the separated flow over an airfoil. A shear layer is
generated by allowing two parallel streams of fluid moving at different speeds (initially
separated by a thin interface) to come together. The flow formed as a result of the two
fluids merging is the shear layer. A schematic of the geometry of a plane shear layer is
shown in Figure 1.

Consider modelling the combustor as a plane shear layer, where one fluid is the
fuel and the other fluid is the oxidizer. If, for example, mixing of the fuel and oxidizer
in the combustor could be optimized, less unburned fuel would be exhausted from the
combustor, resulting in a more efficient combustion process. Like the combustor, many
other industrial applications would benefit from a more detailed understanding of shear
layer dynamics and the mixing process in turbulent shear layers.

Previous studies on shear layers have examined many different aspects of natural

and forced shear layers. In the following sections the behavior of the natural layer, the

l

2

forced layer and mixing in the shear layer will be discussed as related to this study.

 

 

———->U2

Figure 1. Schematic of the plane shear layer.

1.1 The natural shear layer

It was first noted by Brown and Roshko (1971, 1974) that in addition to the
numerous small scales, large scale structures were a prominent feature of the natural shear
layer over a wide range of Reynolds numbers. Dimotakis and Brown (1976) showed that
the large scale structures seen by Brown and Roshko persist at Reynolds numbers up to
3 x 10°, based on high speed velocity and downstream distance. It was also noted that the
shear layer dynamics were governed by a global feedback mechanism. This feedback
mechanism suggests that the initiation of the layer at the splitter plate tip is coupled to
the large scale structures located far downstream.

Growth of the natural shear layer was observed to occur as the result of the
amalgamation of neighboring large scale structures by Winant and Browand (1974). This

process, resulting in a single larger structure, termed pairing, results (on average) in the

linear growth of the natural layer.

Many experimentalists have observed large scale, two-dimensional structures in
the natural shear layer, e. g. Browand and Weidman (1976) and Wygnanski, Oster, Fiedler
and Dziomba (1979). It was Miksad (1972), however, who first observed a weaker
longitudinal (streamwise) vortex structure in the (low Reynolds number) natural shear
layer. Later, Konrad (1976) also saw these streamwise structures at much higher
Reynolds numbers, and concluded that the transition to turbulence occurred as a result of
the formation of this secondary flow structure. Breidenthal (1978, 1981) also observed
the streamwise structures and suggested that this structure was, in fact, pairs of counter

rotating vortices.

1.2 The forced shear layer

Forcing refers to the application of a periodic disturbance to a shear layer in an
attempt to modify the behavior of the layer by exciting the natural instabilities of the
flow. Much work has been done on understanding forced shear layers and many different
methods of forcing have been used. Some of the various forcing methods previously used
include: oscillating one or both of the freestream speeds, Ho and Huang (1982), Roberts
and Roshko (1985) and Koochesfahani and M°°Kinnon (1991), oscillating the splitter plate
tip, Oster and Wygnanski (1982), forcing by acoustical methods, Fiedler et al. (1981) and
Zaman and Hussain (1981), spanwise heating on the surface of the splitter plate, Nyggard
and Glezer (1991), and by oscillating an airfoil downstream of the splitter plate,
Koochesfahani and Dimotakis (1989).

Perhaps the most important observation of forced shear layers is the control over

4

the growth rate and the structure of the shear layer by external forcing. It was reported
by Oster and Wygnanski (1982) and Ho and Huang (1982) among others, that low
amplitude two-dimensional periodic forcing can be used, to some extent, to control the
growth and structure of the shear layer. The effect of forcing on the growth of the shear
layer is demonstrated in Figure 2, reproduced from Browand and Ho (1983). In this

Figure shear layer growth is quantified by the local momentum thickness of the shear

 

 

   

 

 

 

 

.25 a . . 4
Browand and Ho
Based on data from:
Hygnansk: and Oster (1980)
20 _ Ho and Wang (1982)
.15 ~
f9 81000- 0.0681
U. Asymptotic Growth
.10 r Rate for thinned
Mixing Layer
d9
— = 0.0341
dz
.05_ - mo and Latino) '
l = Ul-UZ
U1+Uz
o l l 1 l
0 1.0 2.0 3.0 4.0 6.0
131
U

Figure 2. Effect of forcing on the shear layer growth, reprinted from Browand and
Ho (1983).

layer as defined by

+

0:f (Ul'U)(U‘U2)dy.
(U17Uz)2

 

—00

In this figure the independent axis is the Wygnanski-Oster parameter, x" =x A f / Uc,
where x is the distance from the splitter plate tip, it = (U 1—U2)/ (U 1+U2) is the shear layer
growth rate parameter, f is the forcing frequency and Uc = (U 1+ 2) / 2. This parameter
is used to define three distinct regions of the evolution of the forced shear layer.

The first region, called the enhanced growth region, x *< 1, exhibits enhanced
growth rates up to twice that of the natural shear layer. The next region, the frequency
locked region, 1 < x * < 2, shows reduced growth rates when compared to the natural
layer. In this region the structures are observed to be equally spaced, and have passage
frequency equal to the forcing frequency. In the third region, x * > 2, the growth rate

relaxes to the natural growth rate.

1.3 Mixing in the shear layer

A majority of the experimental and computational studies to date have focused on
the effects of forcing on the momentum transport properties of the shear layer such as the
layer growth rate, the resulting velocity and vorticity fields and their fluctuating
components and the Reynolds stresses. By contrast, there have been few detailed studies
devoted specifically to the behavior of the scalar mixing field in forced shear layers.

Throughout this work, mixing will refer to the mixing of species, the kind of mixing that

6

is necessary for chemical reaction and combustion.

Konrad (1976) and Breidenthal (1978, 1981) were among the first to consider
molecular mixing in the shear layer. Konrad (1976) found in a non-reacting gaseous
layer, a rapid increase in the amount of mixed fluid at some distance downstream of the
splitter plate. The increase in mixing was attributed to increased interfacial area between
the two fluids resulting from the development of small-scale motion which were generated
during transition to turbulence.
Dimotakis (1986) also found this dramatic increase in mixing, in chemically reacting
liquid layers, and referred to this as the mixing transition. The mixing transition is shown

in Figure 3, reproduced from Roshko (1990). The velocity ratio r = U1/U2 in Figure 3

is 2.6.

Mixed Fluid Fraction
(arbitrary units)

Figure 3.

Breidenthal (1978, 1981) and Koochesfahani and

 

1.0 l” T
l" KONRAD

 

 

 

 

I
/
/
__ ens
r- A ..
‘ «30 863% @9
0 BREIDENTHAL 0
_ A KOOCHESFAHANI g
uouuo A 0‘00
0 o O O Q) 0
l _1 2
I02 no’ 10‘ AU 5
v

The mixing transition, reprinted from Roshko (1990).

 

 

7

The mixing transition has been shown to have an effect on the mixing transition,
see Figure 4, reproduced from Breidenthal (1978). In Figure 4 the velocity ratio is
defined as r = U2 / U j , the inverse of the notation used in this work. In the current work

the velocity ratio is r = U 1 /U2 = 2.

 

I I

Konrad Sc a 0-7

,/ rs ‘éfso-ss

 

9
on
I
l
l

9
o
I
1

Mixed Fluid Fraction
(arbitrary units)

 

 

 

 

04 r- .-
6 (no 8am?) Q59
0 = 0-38
0.2 ,_ 6&9 O O Oo”
80 0 Do r 0'76
0
O. §3 0 o 0%10 Q) 00 1 l
102 IO’ N).
AUS
V

Figure 4. Effect of the velocity ration on the mixing transition, reprinted
from Breidenthal (1978).

Roberts (1984, 1985) observed the amount of chemical product in a liquid shear
layer using chemically reacting techniques. It was found that the effect of two-
dimensional forcing can significantly alter the amount of chemical product in the layer.
In pre-transitional flows there were large increases in the amount of chemical product in
the frequency locked region. In post-transitional flows increases were observed only in

the very early stages of the enhanced growth region. In the frequency locked region and

8

beyond there was actually less chemical product in the forced layer than in the natural
layer. The technique of Laser Induced Fluorescence (LIF) introduced by Koochesfahani
and Dimotakis ( 1986), was used to measure the chemical product in reacting flows and
the composition of mixed fluid in non-reacting flows.

The effects of periodic oscillation of the high—speed freestream on the composition
of mixed fluid in a non-reacting shear layer were reported by Koochesfahani and
M°°Kinnon (1991). It was found that while the total amount of mixed fluid integrated
across the layer had increased, the amount of mixed fluid per unit width of the layer had,
in fact, remained nearly constant. This suggests that the larger growth rate of a shear
layer forced by a two-dimensional disturbance does not necessarily lead to a more
efficient mixer. In this work efficiency, as related to the mixing of species in a shear
layer, is defined as the amount of mixed fluid per unit width of the layer. It was also
noted that forcing shifted the predominant mixed fluid concentration to higher values, i.e.

larger proportion of high-speed to low-speed fluid.

1.4 Objectives

As stated earlier, there are many different ways to perturb a shear layer. All of
the methods previously discussed impose disturbances on the shear layer by different
mechanisms, each entering the flow in a different way. Surprisingly, the net result on the
growth rate and structure of the shear layer in response to forcing is very similar for all
methods. The goal of the present work is to investigate the influence of the forcing
mechanism on the mixing of species in a two stream shear layer.

In an earlier investigation, the effects of the periodic oscillation of the high-speed

9

freestream on the composition of mixed fluid in a plane shear layer were reported by
Koochesfahani and M°°Kinnon (1991). The results from the oscillating freestream
experiments are compared to a shear layer forced by oscillating an airfoil located
downstream of the splitter plate tip. In addition to comparing the methods of forcing, the
effect of the forcing amplitude and streamwise evolution on the total amount of mixed
fluid in the layer, the amount of mixed fluid in the central portion of the layer and the

composition of mixed fluid will also be examined.

Chapter 2

EXPERIMENTAL FACILITY AND INSTRUMENTATION

All of the data presented in this work were acquired over a two day period, in two
sets of experiments. Each of the two experiments consisted of ten forcing conditions and
covered a different streamwise view of the test section. In this chapter the shear layer

facility, forcing mechanism and diagnostics are described.

2.1 Experimental facility

The experiments were performed in a gravity-driven water shear layer apparatus,
a schematic of the facility is shown in Figure 5. For reference, the inside dimensions of
the test section were 4 (height) x 8 (width) x 36 (length) cm. The system was fed from
two 210 liter reservoirs, one for each the high-speed and low-speed sides of the shear
layer. Water was pumped from the reservoirs to constant head tanks, located
approximately eight feet above the test section. Flow through the test section was
regulated by three valves; one valve in each of the supply lines upstream of the
contraction, and a third valve downstream of the test section.

For these experiments the freestream velocities were set to U 1 = 40 cm/s and U2
= 20 cm/s, yielding a velocity ratio of, r = 2. In this work the flow was examined over
the range 12.5 < x < 20.0 cm, corresponding to 0.28 < x * < 1.78. The Reynolds number,

10

11

Constant Head Tanks

  
    
    

Control Valves

Contraction Test Section

    
   

   
  

  
   
   

Low High
Speed . Speed - Pumps
Reservmr Reservorr

To Drain <

Figure 5. Schematic of the shear layer facility.

Re,31 = AU 51/ v , based on velocity difference and shear layer width, ranges from 4400
< Re,Sl < 6000. This range of Reynolds numbers corresponds to the later stages of the
mixing transition, see Breidenthal (1981) and Figure 4. It is not clear however, if the
presence of the airfoil affects the mixing transition. Due to the constraints of the flow
facility, this is nearly the highest Reynolds number currently attainable. The natural shear
layer roll up frequency f0, at the splitter plate tip was approximately 27 Hz.

An important consideration to flow quality was the development of the boundary
layer on both the upper and lower surfaces of the splitter plate. A series of screens and

straws were used to create a uniform flow through the contraction. Additionally, extreme

12

care was taken during the filling process to expel all air from the system. Any air trapped
under the splitter plate or in the straws may affect the boundary layer growth and the

resulting shear layer.

2.2 Forcing mechanism

Perturbations were introduced into the shear layer by a 15% thick, 2-D airfoil of
chord C = 2 cm, spanning the entire width of the test section b = 8 cm, oscillating
sinusoidally in pitch about the quarter-chord with angle of attack amplitude A and

frequency f. The airfoil was rotated about its quarter-chord point, located midway

 

 

 

 

6.5 cm

 

l1
[\

AL.

 

 

Figure 6. Schematic of the forcing arrangement.

13

between the upper and lower surfaces of the test section and 6.5 cm downstream of the
splitter plate tip. A schematic of the forcing mechanism is shown in Figure 6. The
placement of the airfoil was determined by the physical constraints of the airfoil
assembly. Movement of the airfoil was achieved by an electro—magnetic coil vibrator
(Vibrations Test Systems VG 100-8). To improve the sinusoidal motions of the airfoil
a P-D feedback control system was used to match the airfoil position to a command input.

The angle of attack of the airfoil varied between -A and +A, with a mean angle
of attack approximately zero. In this work an unforced case and nine forced cases were
examined. In the unforced case the airfoil was stationery at zero degrees angle of attack.
The forced cases included three frequencies at three amplitudes each; the combinations

examined are shown in Table 1.

Table 1. Forcing frequency-amplitude combinations.

 

 

f (HZ) A (degrees)
2 2, 4, 8
4 2, 4, 6
8 2, 4, 6

 

2.3 Diagnostics

The mixing field was measured using Laser Induced Fluorescence (LIF) in the
passive scalar (non-reacting) mode. This technique utilizes a passive scalar containment,
such as a fluorescent dye that is initially mixed with one of the freestream fluids. In this

case the low—speed fluid was marked with the fluorescent dye disodium fluorescein. The

 

l4

dye is then diluted as a result of mixing with fluid from the high-speed stream. By
recording the fluorescence intensity issuing from a sample volume as a function of time,
a quantitative measurement of the dye concentration, and therefore the relative
concentration of high-speed to low-speed fluid within the volume may be made. The

instantaneous dye concentration Cd in the sample volume is defined by

V2

9
+122

 

0 V]

where C d is the freestream (low-speed) dye concentration, v1 and v2 are the volume of
fluid from the high- and low-speed freestreams, respectively, within the sample volume.

Throughout this work the concentration é will be normalized as

In terms of the high—speed volume fraction, concentration may be written as

a — V1
(V1+ V2) -
Therefore, pure fluid from the low- and high- speed sides have concentrations E, = 0 and
Q = 1, respectively.
The LIF measurements were carried out over a plane defined by a laser sheet,

approximately 0.5 mm thick, aligned along the flow direction at the shear layer mid-span

location. The laser sheet was formed by passing a 3 watt beam from a Lexel model 95

15

argon-ion laser through a converging lens and a spherical lens, see the schematic of the
optical setup in Figure 7. The fluorescence intensity was imaged onto an electronically

shuttered CCD (NEC TI-24A) camera (512 x 480 pixels) operating at standard video rate

Argon-Ion laser beam

converging lens

cylindrical lens

laser sheet

 

/

 

17 cm /

 

7V

Figure 7. Schematic of the optical arrangement.

 

16

(30 frames/sec) with an exposure time of 2 msec. Images were then digitized to 8 bits
and captured onto hard disk in real time by an image acquisition system (Recognition
Concepts, Inc., TRAPIX-5500). For each of the ten forcing conditions, 256 consecutive
LIF images (512 fields) were acquired, corresponding to 32, 64, and 128 large scale
structures passing the field of View for forcing frequencies off = 2, 4, 8 Hz, respectively.

The 4 cm test section height was imaged onto 200 of 480 vertical pixels; 512
pixels correspond to approximately 10 cm in the streamwise direction. The resulting
resolution is approximately 200 x 200 pm. It should be noted that at this resolution the
smallest expected diffusion (Batchelor) scales, AD, are not resolved. According to Miller
and Dimotakis (1991), the smallest viscous scale, Av, and the Kolmogorov scale, AK 2
51 Re ‘3”, are related by Av ~ 25 AK. The smallest diffusion scale AD is another factor
of Sc 1’2 smaller than the smallest viscous scale, resulting in AD z )1 K for water (Sc 2
600). Over the range of parameters examined in this study the smallest diffusion scales,

it

D, are estimated to be about 41 um, approximately 5 times smaller than the spatial
resolution of the current memeasurement.

A curious limitation of this technique is that the measurements always over—
estimate the amount of mixed fluid. Unmixed low- and high—speed fluids may
simultaneously exist within the measurement resolution and appear as if they were
actually mixed at that ratio. The results presented in this work are, therefore, not intended
to provide an absolute measure of the extent of molecular mixing. The primary focus of
these results is to characterize the relative changes in the mixing field in a forced layer

compared to the unforced layer. Such comparisons are believed to be warranted since the

resolution, compared to the smallest diffusion scale, is the same for all of the

17

measurements presented in this work.

2.4 Data reduction

The first, and perhaps the most time consuming step in the data reduction process,
is the normalization of the images. The images were first processed to remove the
Gaussian distribution of the laser sheet and nonuniform pixel response of the CCD
camera. A calibration was performed at the start of the experiment to capture these
effects. During the calibration the test section was flooded with low-speed fluid mixed
with the dye, images were recorded and then averaged to produce a calibration image.
The nonuniformities in the pre-processed images were then removed on a pixel by pixel

basis according to

Icon : —I— ’
[cal

where I is the corrected intensity, I is the instantaneous image intensity and 1“,, is the

C017

intensity from the calibration image. Concentration may then be computed as

 

g _ 1 ICOIT
I ’

C OIT

where Icon is the corrected freestream dye intensity. This expression is simply a
conversion from the measured quantity, fluorescence intensity to dye concentration.

The absorption coefficient a = eoCd, where 50 is the dye molar absorption coefficient

and Cd is the molar dye concentration, was measured by Koochesfahani (1984) to be

18

0.157 cm‘1 at a dye concentration of Cd = 10'5M. Attenuation of the fluorescence intensity
at a given depth y is then given by I = I0 e “W , where I0 is the initial intensity. In the
current work, the dye concentration is approximately 2.5x10’7M; a factor of 40 smaller
than in the work of Koochesfahani ( 1984). In the event that the test section is completely
filled with dye, the worst possible case, the fluorescence intensity at the bottom of the test
section would have been attenuated by only 1.6%. The effect of attenuation is much
smaller than the overall signal-to-noise ratio of the experiment 8 = 4.5%.

From the normalized images the probability density function (pdf) of the
concentration field can be constructed at any of the measurement points in the x-y plane.
The pdf of concentration 5 as a function of position y in the layer, written p (E, y), was
computed at six different streamwise locations. The pdf p (E, y) is the basis for all of
the computed quantities in this study. The ideal pdf would have delta functions at E =
0 and 1 representing pure fluid from the low- and high-speed freestreams, respectively.
In practice, the delta functions have a finite width 8 which is the result of the overall
signal-to-noise ratio of the experiment. In this work 8 = 0.045, therefore concentrations
in the range 0 S E < e and 1-8 < E, S 1 are considered to be pure unmixed fluid from the
low- and high-speed freestreams, respectively. Likewise, concentrations within the range
8 < g < 1-8 are considered mixed fluid.

The first quantity computed from p (E, y) is the average concentration E which

is found by integrating over all concentrations at a given position, as defined by

Em = f amends.
0

19

To characterize the effect of forcing on the mixing field the probability of occurrence of

unmixed fluid from either the low- or high-speed freestreams can be computed by
poo) = f p (a, we,
0

1

p10) = f panda.

1..
Similarly, the probability of occurrence of mixed fluid is computed by integrating over
the range of concentrations defining mixed fluid as given by

1-5

pm(y)= f p(€.y)d€.

8

It should be noted that pm is the probability of occurrence of mixed fluid of all
concentrations; a value of less than one implies the presence of unmixed fluid.

The shear layer width 81 may be computed based on the total mixed fluid
probability curve at a given downstream location. The local width is defined in a manner
similar to boundary layer thickness. Recall, the boundary layer thickness is the height
above a surface where the local velocity is 99% of the free stream velocity. The shear
layer width 51 is similarly defined as the distance between the two points where the
probability of mixed fluid is 1% of the maximum value of pm (y). Koochesfahani and
Dimotakis (1986) reported that the visual width of the shear layer 5”,. is approximately
equal to 5,.

A measure of the total amount of mixed fluid (at all concentrations) in the layer,

20

the mixed fluid thickness, is defined by

h '1—8 h

8... = f f p(5,y)dady = pm (My.

-h e -h
Note that the mixed fluid thickness is simply the area under the total mixed fluid
probability curve. The quantity 8m/81, termed the mixed fluid fraction, is used to
quantify the amount of mixed fluid per unit width of the shear layer.

It has been shown by Koochesfahani and M°°Kinnon (1991) that the distribution
of the mixed fluid composition is nearly invariant across the entire width of the layer, and

therefore may be characterized by an average pdf. The average pdf is defined by

+h
P(€) = i p(E,y)dy.

—h
where h is the test section half width.

A common measure of shear layer growth is the momentum thickness defined by

+00

(Urinal—U.) d
(U? Uz)2 °

 

In this work no velocity measurements were made, therefore the momentum thickness
cannot be calculated. Another integral thickness, called the concentration thickness, may
be defined in a manner analogous to the momentum thickness, based on the average

concentration, defined as

21

 

Recall that the subscripts 1 and 2 in the above equation refer to the average concentration
of the high- and low—speed freestreams, respectively. The average concentration of the
high-speed freestream E1 = 1 and for the low-speed freestream 22 = 0. The

concentration thickness is then reduced to

The relationship between the shear layer width 81 and the concentration thickness 0P. will

be examined in section 3.4.

Chapter 3

RESULTS AND DISCUSSION

3.1 Flow visualization results

Figures 8 - 12 illustrate the influence of the forcing frequency and amplitude on
the structure and growth of the shear layer. In all of the images the flow is from right
to left with the high—speed stream on top. The time between images At is 1/30 sec, (every
other field) with time increasing from top to bottom. Each image, composed of 90,000
pixels (450 in x and 200 in y), spans the entire height of the test section and the range
12 < x < 22 cm (right to left). After processing, the images were pseudo-colored to
simulate a chemical reaction. Pure fluid from either of the freestreams was assigned to
black. Fluid composed of mainly low-speed fluid, I; z 5, were assigned shades of blue;
likewise nearly pure high-speed fluid, g z 1-8, were colored shades of red (recall pure
low-speed fluid has concentration 0 S E, S e, high-speed fluid l-e S é S 1).

Two different time series of images from the unforced layer are presented in
Figure 8. These clearly show the random nature of the unforced layer. In Figure 8 (a)
a fairly well defined large structure may be seen passing the field of view, while some
time later in Figure 8 (b), there are no well defined large structures apparent. It should

be noted that the unforced shear layer described here is not the same as the natural layer

22

23

due to the presence of the airfoil. Differences between the unforced and natural shear
layers will be discussed in Section 3.6.

Figures 9 - 12 display the effect of the forcing amplitude A on the structure and
growth of the shear layer forced at f = 2, 4 and 8 Hz, respectively. For the case forced
at f = 2 Hz, A = 2°, see Figure 9 (a), there is very little if any Visible effect on the
structure and growth rate of the layer, when compared to the unforced case. With
increasing amplitude, Figures 9 (b) and (c), the layer growth rate increases resulting in
larger vortical structures and shear layer width at a given downstream location x. As a
result of the larger layer width mixed-fluid can also be seen over a larger portion of the
test section. Quantitative descriptions of the shear layer width and the amount of mixed
fluid will be given in section 3.3. There are some periodic structures in the flow,
however they are not very well defined. The location where the structures are fully
formed is the center of the frequency locked region, x" = 1.5; for f = 2 Hz this
corresponds to x z 68 cm, which lies well beyond the end of the test section. See section
1.2 for description of the nondimensional streamwise coordinate x *. Moving farther
downstream, if possible, would not allow the structures to further develop because of the
limiting height of the test section. The lower wall of the test section may be seen to
influence the structures in Figure 9 (c). In the images, the nondimensional streamwise
coordinate spans the range, 0.26 < x *< 0.52, which lies well within the enhanced growth
region.

In Figure 10 the structure of the shear layer is shown as a function of the forcing
amplitude A when forced at f = 4 Hz. At this frequency the nondimensional streamwise

coordinate spans the end of the enhanced growth rate region, 0.52 < x *< 1.02. Again,

24

 

Figure 8. Time series of the unforced shear layer. (a) and (b) are not consecutive in
time, flow is from right to left, At = 1/30 sec.

 

(a)A=2° (b)A=4° (c)A =8°

 

Figure 9. Effect of the forcing amplitude on the structure of the shear layer forced at
f = 2 Hz, flow is from right to left, At = 1/30 sec.

26

 

(a)A =2° (b)A=4° (c)A =6°

 

§=0 )1 §=1

Figure 10. Effect of the forcing amplitude on the structure of the shear layer forced
atf: 4 Hz, flow is from right to left. At = 1/30 sec.

27

 

 

(a)A=2° (b)A=4o (c)A =6°
5. = 0 E = 1
Figure 11. Effect of the forcing amplitude on the structure of the shear layer forced

atf= 8 Hz. flow is from right to left, At = 1/30 sec.

28

 

Figure 12. Effect of the forcing frequency on the structure of the shear layer forced
at A = 4°, flow is from right to left, At = 1/30 sec.

29

it may be seen that as the amplitude increases the layer growth rate and the amount of
mixed fluid within the layer also increases. In this case the structures are more well
defined than the case forced at 2 Hz. At the largest amplitude the structures are, again,
approaching the width of the test section.

The cases forced at f = 8 Hz, shown in Figure 11, span the frequency locked
region (1.04 <x *< 2.04) characterized by an array of non-interacting, equally spaced
vortices and reduced growth rate. Even at the smallest amplitude A = 2°, the structures
are very well defined. As in the previous cases, the effect of amplitude is shown to result
in an increase in the layer width and the amount of mixed fluid within the layer. In all
of these cases, the increase in shear layer growth rate, in response to low frequency
forcing (relative to the natural frequency, f0 = 27 Hz), culminating in the formation of
large vortical structures is consistent with previous shear layer studies eg. Fiedler et al.
(1981), Oster and Wygnanski (1982), Ho and Huang (1982), Koochesfahani and
Dimotakis (1989). For completeness, Figure 12 is included to show the effect of the
forcing frequency on the structure of the shear layer when forced at a fixed amplitude of
A = 4°. Note as the frequency is increased, the excursions of low-speed fluid to the upper
side, and high-speed fluid to the lower side of the shear layer becomes deeper. This

excursion of fluid is often referred to as an entrainment tongue.

3.2 Transverse profiles of 5, pm, p0 and p1
The quantities computed from the probability density function of the high—speed
fluid volume fraction p (E , y) are presented in this section. The plots are divided into two

sections. The first section will present the evolution of the various computed quantities

30

as a function of streamwise location x, at a fixed forcing frequency and amplitude. The
second will show the amplitude dependence at a fixed downstream location x. Data were
computed at 6 uniformly spaced streamwise locations spanning the range 0.28 S x' S 1.78

(12.5 S x S 20.0 cm).

3.2.1 Streamwise dependence

Transverse profiles of the average concentration E, the total mixed fluid probability
pm (y) and unmixed fluid probabilities, p0 (y) and p1 (y) for the unforced shear layer are
shown in Figure 13. In general, the results for the unforced layer are very similar in
shape to the results of the natural layer reported by Koochesfahani and Dimotakis (1986)
and Koochesfahani and M°°Kinnon (1991). As previously discussed, differences in the
shear layer width and the amount of mixed fluid between the unforced and natural layers
will be quantified in section 3.6.

The most obvious feature of the unforced layer is the increase in the layer width
with increasing downstream distance as seen in the total mixed fluid probability curves
displayed in Figure 13 (b). Increasing shear layer width may also be inferred from
outward movement of the probability of pure low- and high-speed curves, see Figure 13
(c) and (d). A feature unique to the unforced layer is the nearly constant value of the
peak of the mixed fluid probability curve (pm)m= 0.95 with increasing downstream
distance. Recall that a value of pm less than unity implies the presence of unmixed fluid;
therefore roughly 5% of the fluid in the center of the unforced layer is unmixed. It will
be shown later, that there can be a significant streamwise variation in (pm) in the

max

forced cases.

31

 

 

1.0

0.8

 

 

 

 

 

  

41111111111111

 

 

O ' ‘ .‘
~20 -10 0 10
y (mm)
(a) Average concentration (b) Total mixed fluid probability

 

 

 

 

 

 

 

 

 

l l l l l l I I l 1.0 I l I l I l l l l l l l l
—. 0 8 _ {fl §X _
. - C177: _
r 0.6 r ' -' r
._ it
o. — :l, I _
— 0.4 — ' —
_ gr -
a j. r a
' 0 l 1 L 1 1 l 1 1 1 1
10 20 -20 -10 0 10 20
3’ (mm) y (Inn!)
(c) Pure low-speed fluid probability ((1) Pure high-speed fluid probability

0 x: 12.5, Ox: 14.0, V x=15.5, Ox: 17.0, I x=18.5, A x: 20.0 cm

Figure 13. Transverse profiles for the unforced shear layer as a function of the
downstream distance x.

32

 

 

 

 

 

 

 

 

 

 

1.01"'1'f'fil'llv‘u 1.OIIII]IIII]1fiIT]IfII
0.8 —
0.6 '
my _
0.4 r
0.2 r
0 11111111111111
-20 -10 0 10
y(mm) y(mm)
(a) Average concentration (b) Total mixed fluid probability

 

 

 

 

 

 

 

 

 

 

 

I I I I I I 1.0 I l I I I I l I I I I I
.1 .- Iéig ‘a —.
a 0.8 r ' r
‘ l
r 0.6 r -
- g: _ _
r 0.4 r r
- - l
r 0.2 r a
- t 1
1 O 1 1 l
10 20 -20 20
y (Hun) y (Inn!)
(c) Pure low-speed fluid probability ((1) Pure high-speed fluid probability
0 x* = 0.28, D x* = 0.31, V x’“ = 0.34, 0 x* = 0.37, l x‘ = 0.41, A xt = 0.44
Figure 14. Transverse profiles for the shear layer forced at f = 2 Hz, A = 2° as a

function of the nondimensional downstream distance x'i.

33

 

 

     

 

 

 

 

   

 

 

 

 

 

 

 

 

 

 

 

 

1.0 I 1.0 I I I I I I I I I I I I I I I I I I I
0.8
0.6
MU"
0.4
0.2
O I l
-20
(a) Average concentration
1.0 I I I I I I I I I I I I I I
0.8 —
0.6 —
O
Q _
0.4 -
0.2 —
O I‘L. ‘1 0 1111111111
-20 -10 0 10 20 -20 -10 0 10 20
y (Hun) y (turn)
(c) Pure low-speed fluid probability ((1) Pure high-speed fluid probability
0 x" = 0.28, D x” = 0.31, V x'" = 0.34, 0 x" = 0.37, I x" = 0.41, A x” = 0.44
Figure 15. Transverse profiles for the shear layer forced at f = 2 Hz, A = 4° as a

function of the nondimensional downstream distance x'“.

34

 

 

 

4%. 94°

'1
x36
1. _
A"
I" ‘
"1 02..— _
o o \\
A
A
Av — -1
4'," é
,o ’
01111111111111111 0 1111111111111

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

-20 -10 0 10 20 -20 -10 0 10 20
y (m) y (m)
(a) Average concentration (b) Total mixed fluid probability
1.0 I I I I I I I I I I f I I I I I I I 1.0 I I I I I I I I I r I I I I
0.8
0.6
a?
0.4
0.2
0
-20 20
y (HUD) 3’ (mm)
(c) Pure low-speed fluid probability ((1) Pure high-speed fluid probability
0 x' = 0.28, 1: x’ = 0.31, v x’ = 0.34, o x‘ = 0.37, I x" = 0.41, A x" = 0.44
Figure 16. Transverse profiles for the shear layer forced at f = 2 Hz, A = 8° as a

function of the nondimensional downstream distance x'.

 

 

 

1.0

0.8 *

 

Figure 17.

 

11111111111111

0 10
y (m)

20

(a) Average concentration

 

I I I I
§\ gi‘ _
\
I
\

\
I
\
I
I
I
‘ I
\
\
I
\

I I I I I I I I I I I

lit.

  

 

 

10
y (m)

(c) Pure low-speed fluid probability

35

 

1.0

 

 

 

I I I I I

I I I I I I I I I I I I I I

l

1 4 M 4

l

l

A;

 

 

y (m)
(b) Total mixed fluid probability

 

 

IIIIIIIIIIIIII
1!
’1
I
1
1

 

y (m)

((1) Pure high-speed fluid probability

0 x* = 0.56, D x" = 0.62, V x" = 0.68, O x* = 0.75, I x’ = 0.82, A x* = 0.88

Transverse profiles for the shear layer forced at f = 4 Hz, A = 2° as a

function of the nondimensional downstream distance x’“.

36

 

 

 

 

 

 

 

 

 

 

1.0 ' I I I I I I I I I I I I I | I I I 1
0.8 r —
0.6 — —
0.4 — —
0.2 _ f’l I “1 ‘ 1 ‘
.- I I, ‘. \ 1 _
O figj 1 1 1 1 n 1 1 1 1 1 w
-20 -10 0 10 20
y (m)
(a) Average concentration (b) Total mixed fluid probability

 

 

1.0 I I I I I I I I I I I I I I I I 1.0 I I I I I I I I I I I T I I
.. ‘ \ I _
I
l
\ ‘ ‘
\
I
O O

 

 

 

 

 

 

 

 

-20 -10 0 10 20 -20 -10 0 10 20

y (Hun) y (Hun)
(c) Pure low-speed fluid probability ((1) Pure high-speed fluid probability

0 x’" = 0.56, D x* = 0.62, V x" = 0.68, O x* = 0.75, I x‘ = 0.82, A x* = 0.88

Figure 18. Transverse profiles for the shear layer forced at f = 4 Hz, A = 4° as a
function of the nondimensional downstream distance x'".

 

37

 

1.0 I I I I I I I I I I I I I I I I I I I

 

 

 

 

 

 

 

 

télgg . —
0 1111111111111111 O 111111111111%

-20 -10 0 10 20 -20 -10 0 10 20
y (m) y (m)
(a) Average concentration (b) Total mixed fluid probability

 

 

1.0 I I I I I I I I I I I I I‘I I I 1.0 I I I I I I I I I I I I I T
\

 

 

 

 

 

 

 

(c) Pure low-speed fluid probability (d) Pure high-speed fluid probability

0 x' = 0.56, D x' = 0.62, V x" = 0.68, 0 x* = 0.75, I x‘" = 0.82, A x* = 0.88

Figure 19. Transverse profiles for the shear layer forced at f = 4 Hz, A = 6° as a
function of the nondimensional downstream distance x".

38

 

 

 

 

 

 

 

 

 

 

 

 

  

 

 

 

 

 

 

 

 

1.0 1.0 ' ' . ' l I . I I I I I . ' I ' I . .
0.8 r 0.8 r _
0.6 — 0.6 — r
1w - a? t _
0.4 — 0,4 — _
0.2 — 0.2 ~ _
”' I . F '1
O "i" . ' ‘ 1 1 1 1 I 1 1 1 1 1 1 1 1 1 0
-20 ~10 0 10 20 ~20 20
y (man)
(a) Average concentration (b) Total mixed fluid probability
1.0 m I I I I I T T I I I I I I I 1.0 I I I I I I I I I I I I I I
_ w ‘8? _ _ $137 _
0.8 - 1‘ — 0.8 - ', _
_ Ill? _ . Hf _
\ \‘ I 1
0.6 r r 0.6 r a
a? — l , . o: — «
0.4 r r 0.4 r r
0.2 — r 0.2 r —
’ ‘ ' l
O 1 1 1 L l 1 1 1 ‘ 0 ma L1 1 1 L 1 1 1 1 1
-20 -10 10 20 -20 -10 0 10 20
y (mm
(c) Pure low-speed fluid probability ((1) Pure high-speed fluid probability
0 x" = 1.11, D x* = 1.24, V x* = 1.38, 0 x“ = 1.51, I x* =1.64, A x‘ =1.78
Figure 20. Transverse profiles for the shear layer forced at f = 8 Hz, A = 2° as a

function of the nondimensional downstream distance x".

39

1.0 I I I I I I T I I I I I I 1.0 I I I I I I I I I I I I I I I I I I I
I
I
c ‘ Ait.‘ ‘ ""‘ [I
L I ‘
LI , A I‘ a “
1 v ' HA I \‘_
‘5
I -

 

 

‘ 0.8 —

 
   
   

- 0.2 W

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

0 1 l 1 1 1 1 l 1 1 1 1
-20 -10 0 10 20 -20
y (nun) y (Inna)
(a) Average concentration (b) Total mixed fluid probability
1.0 II I I I I I I I I I I I l I I I 1.0 I I I I I I I I I I I I l I
IE9 ~ ~ Ilpv’ —
0.8 _ 1 . .1 4%
0.6 — ‘
O
a. _
0.4 — —
0.2 — _
O 1 1 r 1 1 1 1 | 1 1 1 1 l 1 1 1
—20 -10 0 10 20
y (m)
(c) Pure low-speed fluid probability ((1) Pure high-speed fluid probability
0 )6“ =1.11, D x“ =1.24,V x’“ =1.38,0 x' = 1.51, I x’“ =1.64, A x" =1.78
Figure 21. Transverse profiles for the shear layer forced at f = 8 Hz, A = 4° as a

function of the nondimensional downstream distance x’.

40

 

 

 

 

 

 

 

 

 

 

 

 

1.0 1.0 I I I ITI I I I I I I I I I I I I I
0.8 " 0.8 —
0.6 _ 0.6 —
1w _ DEE _
0.4 “ 0.4 —
I. _
0.2 r 0.2 —
~ _ I
O N ;a 1 l 1 1 1 1 l 1 1 1 1 l 1 1 1 1 0 l 1 1 1 1 l 1 1 1 1 I
-20 -10 0 10 20 -20 -10 0 10 20
y(mm) y(mm)
(a) Average concentration (b) Total mixed fluid probability
1.0 \IqIIIIIIIIIIIIffIIJ I'OIIIIIIIIIIIIIIW
" MI " $9311 ‘
08* If): a 08— I" —
u \| b‘ . WI
_ 1‘ ‘ " '1 VI _

 

 

 

 

 

 

 

- i
t _
1 1 1 1 l 1 1 1 1 l 1 1 1 1
-20 -10 0 10 20
y (mn)
(c) Pure low-speed fluid probability (d) Pure high-speed fluid probability
0 x”! =1.11, D x* = 1.24, V x* =1.38, 0 )6“ =1.51, l x" =1.64, A x‘ =1.78

Figure 21. Transverse profiles for the shear layer forced at f = 8 Hz, A = 6° as a

function of the nondimensional downstream distance x‘.

41

Results for the forced cases are displayed in Figures 14 - 22. A noteworthy
feature of the forced shear layer is the severe distortion of the average concentration
profiles EU»). The development of a 'flat region' in the center of the 80)) profile is
characteristic of the passage of well mixed vortex cores, previously noted by Wygnanski,
Oster and Fiedler (1979) and Koochesfahani and M°°Kinnon (1991). This 'flat region' is
seen developing in the cases forced at f = 4 Hz and is fully developed in the 8 Hz cases.

Similar to the unforced layer, the forced cases all show increasing shear layer
width as downstream distance increases. Correspondingly, mixed fluid is found over a
wider portion of the test section width. There is however, a reduction in the amount of
mixed fluid (pm)max in the middle of the layer as the entrainment tongues become more
prominent. For the cases forced at f = 2 Hz, there is a slight variation in (pm)max as a
function of x, similar to the unforced layer. However, with increasing amplitude there is
a noted decrease in (pm)m, see Figures 14 - 16. When forced at f = 4 Hz, A = 2°, see
Figure 17, there is a continual decrease in the amount of mixed fluid in the center of the
layer. At the larger amplitudes A = 4° and 6° the amount of mixed fluid continually
decreases except at x = 20.0 cm where a slight increase in (pm)max is noted, see Figures
18 and 19 (b). The cases forced at f = 8 Hz show markedly different behavior than the
previous cases. When forced at f = 8 Hz, A = 2° (pm)m initially decreases then
continually increases; while (pm)max continually increases at the higher amplitudes.

The reduction in the amount of mixed fluid in the central portion of the layer is
accompanied by an increase in the amount of the unmixed fluid. In all forced cases there

is more pure high-speed fluid than pure low-speed fluid in the center of the layer. This

is most obvious in the cases forced at f = 4, 8 Hz, see Figures 17 - 22 (c) and (d). The

42

unequal amounts of unmixed fluid in the center of the layer are related to the inherent
asymmetry of the shear layer.

It was previously noted in the flow visualization images of Figures 8 - 12 that
some of the forced cases were influenced by the finite height of the test section. This is
also seen in the total mixed fluid probability curves in Figures 16 (b) (f = 2 Hz, A = 8°)

and 19 (b) (f = 4 Hz, A = 6°) as nonzero values of pm at y = -20 cm.

3.2.2 Forcing amplitude dependence

In the previous section the streamwise dependence of two-dimensional forcing was
noted for the various forcing conditions. In this section the dependence on the forcing
amplitude will be shown at a fixed downstream location. In Figures 23 - 25, transverse
profiles of the average concentration E, the total mixed fluid probability pm (y) and
unmixed fluid probabilities, p0 (y) and p10») are shown as a function of the forcing
amplitude at x = 20.0 cm. The unforced case is included for comparison. Figures for the
remaining measurement locations are provided in Appendix A.

The 'flat region' of the average concentration profiles associated with the passage
of well mixed vortex cores is not present in the 2 Hz cases because the vortices are not
fully formed at this location. There is almost no difference in the average concentration
profiles between the unforced case and the cases forced at amplitudes A = 2° and 4°, see
Figure 22 (a). Well mixed vortex cores are associated with the well defined structures
of the frequency locked region seen in the cases forced at f = 4 and 8 Hz. The
developing 'flat region' may be seen in the 4 Hz cases and a very well defined 'flat region'

is observed in the f = 8 Hz cases, see Figures 24 and 25 (a).

 

 

 

 

 

 

 

 

 

 

 

43
1'0 I I I I I I I I I I I I I I :3. 1.0 I I I I I I I I I I I l I I I IWI 77
0.8 — ~
0.6 r -
ILU‘ _ _
d. I
0.4 - 7 -
.5 V0
_ ,I 4
f v:
0.2 "' ’, Vg ‘
. n‘
_ ,1‘ , -
0 £1 1 1 1 1 I; 1 1 1 l 1 1 1 1
-20 -10 0 10 20 -20 -10 0 10 20
y (mm) y (m)
(a) Average concentration (b) Total mixed fluid probability
1.0 ‘ I I I I I I I I I I I I I I I I I 1.0 I I I I I I I I I T I I I I I
_ b g 4 ~ g ’ r
\‘ \ {a ' ’7
0.8 — '1 V e 4 0.8 r -

\ I
(a _ _ , -

 

 

 

 

l l

10 20 -20 -10 O 10 2O

 

 

 

-20 -1O

 

(c) Pure low-speed fluid probability . ((1) Pure high-speed fluid probability

0 Unforced case, I: A = 2°, V A = 4°, 0 A = 8°

Figure 23. Transverse profiles for the shear layer forced at f = 2 Hz as a function of
amplitude A, at x'" = 0.44 (x = 20.0 cm), compared to the unforced case.

(a)
1.0 I . I I . I I I I
”KW,
0.8 _ \IIQ‘IQ
' IIIg
0.6 _ “I? (I:
a? _ A:
0.4 —
0.2 —
O 1
-20
(c) Pure
Figure 24.

 

 

 

 

Average concentration

 

 

1.0

I I I I I I I I

 

 

low-speed fluid probability

 

 

 

 

 

(b) Total mixed fluid probability

 

 

I I I I I I I

I I I I I I I I

     

o Unforced case, 1:1 A = 2°, V A = 4°, 0 A = 6°

air

\

 

(d) Pure high-speed fluid probability

Transverse profiles for the shear layer forced at f = 4 Hz as a function of
amplitude A, at x" = 0.88 (x = 20.0 cm), compared to the unforced case.

 

 

 

 

 

2O

 

 

 

 

45

E
0..

 

 

  

 

 

 

 

 

 

 

1.0 ' I ' ' I I I I I I I I I I l I 1 1 r
0.8 '— —
0.6 c —
0.4 T .V/ ' 1': wk _
~ 1’ / 5° [III _
/ , , ‘ P
0 2 — y’ 959 “ W —
. ' I; ’ ,0 1‘\\‘
- $198 8% -
0 w 1 l 1 1 1 1 l 1 1 1 1 I 1
-20 -10 0 10 20
y (m)
(b) Total mixed fluid probability
1.0 I I I ' I I I I I l I F I I ]fi
20

 

((1) Pure high-speed fluid probability

0 Unforced case, 1:] A = 2°, V A = 4°, 0 A = 6°

1.0
0.8 _
0.6 _ JV '
”LP _ #¢ ¢' a
04 — 17¢ 0 —
. :7 (5
_ cl I4 . .
,v' 0
1’?! F5 ¢
0.2 1); E25 ‘9,

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-20 -10 0 10

MM)
(a) Average concentration
1.0 IIIIIIIIIIIIIIIII
u Q¥$ ‘Q
~ 15%
0.8 - I. “
0.6 r
O
Q _
0.4 —
0.2 —
0
-20
(c) Pure low-speed fluid probability
Figure 25.

Transverse profiles for the shear layer forced at f = 8 Hz as a function of

amplitude A, at x’" = 1.78 (x = 20.0 cm), compared to the unforced case.

46

In all of these cases, the width of the layer is larger than the unforced case and
increases with increasing amplitude. In this sense the effect of the forcing amplitude on
the shear layer width is similar to increasing the downstream distance. The cases forced
at f = 2 and 4 Hz show a reduction in (pm)mam with increasing amplitude. Coinciding
with the reduction in the amount of mixed fluid in the center of the layer there is an
increase in the amounts of pure fluids from both streams. Note that there is in general
significantly more high—speed fluid in the center and on the lower side than low-speed
fluid in the center and on the upper side. Again, this is related to the inherent
entrainment asymmetry of the shear layer. The cases forced at f = 8 Hz exhibit a similar
behavior. Unlike the 2 and 4 Hz cases, however, there is no dramatic reduction in
(pm)m as a function of amplitude for the 8 Hz cases at this x. In this case, (pm)max is

nearly constant at this location (x = 20.0 cm). This is believed to be connected to the

saturation of the large structures in the frequency locked region.

3.3 Streamwise variation of 51, 8 (Sm/51 and (pm)

m ’ max

The detailed profiles of the previous section can be used to compute the shear
layer width 51 and other mixing quantities such as the mixed-fluid thickness 8m, mixed
fluid fraction 5m/51 and the amount of fluid in the center of the layer (pm)mx. These
results are discussed in this section.

In Figures 26 (a) - (c) the shear layer width 81, is plotted versus x as a function
of amplitude for the forcing frequencies f = 2, 4 and 8 Hz, respectively. The unforced
layer is noted to have linear growth rate. From this Figure, it may also be seen that the

width of the forced layer is greater than the unforced layer and increases with downstream

47

 

40 I I I I I I I I I I I I I I I I
— J
.— -I
_ (a)f= 2 Hz a
35 — .A ---------- 0 "
>—- ”/ d

"/
A ” ______ f", -
E _ x. ’’’’’’ V _
/ II,
v 30 "- ’/’ ’V" ,,,, 9" fig _
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'— ./ //’V , 'jlgf/ -
— /// ’3' """ 'fi
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2 /’ MEI/w“

5 _ ./ -0" —
— V/// /’,/.—"'x a
"' 4:18," ‘
.. g/Zé"- _
- -I

 

 

 

12.5 14.0 15.5 17.0 18.5 20.0

x (cm)

0 Unforced case, B A = 2°, V A = 4°, 0 A = 8°

 

40 I I I I I I I I I I I I I I r I
— —I
— —I
b (b)f= 4 HZ ~
I /-— -------- " .
35 V "/x/ ”’47 “
._ ’//., /’V""—-- ._

‘_/ ///

A I— ”/’ //V —

,’/ // /,/B
v 30 _ ‘,’/ /// /‘B"/' ’,—O —

—1 ._ , ”,
oo /' / ’fl/ - I
._ /,r /V ’/ ”0' ..
,’ /// /,/ _-v I
_ ,/ / /,- d
, / /8 ,v
_ ./ //’V/ ,/’ ’,,e" ’0 q
__ // ,/ _v"’ _
25 _ V// ’/EI/ xxx/O _
I/ y' ’f
— ,/ ”/0 _
_ B//-"' _
G—

 

 

 

20 I 1 1 I 1 1 I 1 l I I 1 I 1 l I
12.5 14.0 15.5 17.0 18.5 20.0

x (cm)

0 Unforced case, D A = 2°, V A = 4°, 0 A 2 6°

Figure 26. Streamwise evolution of the shear layer width 51 as a function of the forcing
amplitude A compared to the unforced case.

48

 

 

 

 

40 I I I j I I I j I I I I I I I 1
~ (c)f= 8 Hz q
35 — _,_-4 ......... a z
— """" .I’ """" ’V d
- A. """ /g/” q
/ /// _
A — / V//
E _ . / //V"-’~—-——' / / / ‘
v 30 _ / /// ”/uB" O _‘
'— - / // ,/B/ / _
w '— / /X’ ,/ / 0 "I
._ ‘ /// [/8/ a —1
— // ,/ ,0 d
V’ ’/ -v""'
I— ,v” _
25 ’/8/ ,-/’O
1.. ./ ’ I, _
_ ,/ -0 _
_ [34--— _
Ga
20 I 1 4 I m # I 1 1 I l 1 I I l I
12.5 14.0 15.5 17.0 18.5 20.0
x (cm)

0 Unforced case, D A = 2°, V A = 4°, 0 A = 6°

Figure 26. Streamwise evolution of the shear layer width 51 as a function of the forcing
amplitude A compared to the unforced case.

distance. The highest amplitude cases all seem to saturate at approximately 35 mm in
width. The forced layer is expected to initially have enhanced growth rate, then taper off
(saturate) and finally resume the natural growth rate, see section 1.2. The growth of the
forced layers may be hindered by the finite size (height) of the test section, 2h 2 40 mm;
or may be reaching the saturation point at x " = l.

The total amount of mixed fluid in terms of the mixed fluid thickness 8m is
plotted in Figure 27. The amplitude dependence for the f = 2 Hz cases are shown in
Figure 27 (a). Only at the largest amplitude A = 8° was there a significant increase,

approximately 11%, in the mixed fluid thickness. There was less than a 2% change in

49

 

 

 

20 I I I I I I I I I I T I I I I
r —1
(a)f= 2 Hz
18 V 7
I. —I
“16 _ ‘
”'/.
O"—
,t—
’/ ‘
v ” ’/’
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°° -/" ,/»1=:B
— /’ i ‘7 —‘
14 '/” /// "—""a'
/,' /:’/,/’,‘.—"
_ ,,,— ‘ ,/<g».—.--—— -
/’ ///’,/.-’
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/” A7,—‘ 4
12 I— .(x Mir/x
/’£"
*- /./"" "
57"
l I l I I l l I

 

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12.5 14.0 15.5 17.0 18.5 20.0

x (cm)

0 Unforced case, D A = 2°, V A = 4°, 0 A = 8°

 

20 I I f I I I I I I I I I I I I

(b)f=4Hz

A 16 — —

,0

I/
— /’ .1
v I
I

V

E x/ ,/:6

(o ////:”
14 _ --/"/ ./ r

, v-
x." / —< '
.—/' 2.2/5"
- --/ 54" —
,,t:,/7.Z"
/- V a,
"/" //"a’/a
12 — ’/’./’:;:/ ..._
/’ / 4"
a’/ ’ ,
/,"

 

 

10 I I 1 I I I I I I I l l I I I I

12.5 14.0 15.5 17.0 18.5 20.0

x (cm)

 

O Unforced case, B A = 2°, V A = 4°, 0 A = 6°

Figure 27. Streamwise evolution of the mixed fluid thickness 5", as a function of the
forcing amplitude A compared to the unforced case.

50

 

 

 

 

20 I I I I I I I I I I I I I I I
(c)f= 8 Hz /0
18 - , / V a
a” /
_ /—’/ //// J
,0’ /V
A 16 — / / ‘
/ // E]
E / // /’
v '— ./ //v / / “
E / /
00 /” // /’E --"/O __
14 _ x // /, -x’“
/ //V ’/’——" —‘0
r- ,/ // /',‘8'"—'— n
1., // —/’ I
x / /, "’
,/ /V /;,./6" _
12 _ /’// ,/ "
’/'// ”VQ/
I- ’//’,:/’ ‘
.;7
10 1 I I I l I I I I I I I I I I
12.5 14.0 15.5 17.0 18.5 20.0
x(cm)

O Unforced case, CI A = 2°, V A = 4°, 0 A = 6°

Figure 27. Streamwise evolution of the mixed fluid thickness 8", as a function of the
forcing amplitude A compared to the unforced case.

the mixed fluid thickness when compared to the unforced shear layer at the two smaller
amplitudes. Small increases, 5% at most, were similarly noted for the cases forced at f
= 4 Hz, see Figure 27 (b). There was, however, a marked increase in the amount of
mixed fluid at all amplitudes when the layer was forced at 8 Hz, Figure 27 (c). The
largest change observed in this study was 29%, when the layer was forced at f = 8 Hz,
A = 6°.

It was noted above that the largest increase in the mixed fluid thickness compared
to the unforced layer was 29%, for the case forced at f = 8 Hz, A = 6°. The shear layer

width in this case was 51 = 35.2 mm, an increase of 19% over the unforced layer (51 =

51

29.6 mm), see Figure 26 (c). One way to quantify the amount of mixing, while taking
into account the increasing width of the layer is to use the mixed fluid fraction 8m /81,
the fraction of the layer width occupied by mixed fluid. 5m/81. In Figure 28, the mixed
fluid fraction is shown as a function of amplitude. In most cases the forced layer has less
mixed fluid per unit width than the unforced layer, and in that sense, is a less efficient
mixer. The exceptions are the cases forced at f = 8 Hz, A : 4° and. 6° at the farthest
downstream locations x = 18.5 and all amplitudes at x = 20.0 cm. The reduction in the
mixed fluid fraction as a result of forcing was similarly reported by Koochesfahani and
M°°Kinnon ( 1991). It is interesting to note that the mixed fluid fraction for the cases

presented here are at best no greater than 0.52; only slightly more than half of the fluid

 

 

 

 

0‘55 I I I I I ' l I I I I I l I I l

L (a)f= 2 Hz _

I ._

-»O ______ _

0.50 I: / /B\—;:::§:—-—-—---—— ———— 0. ‘‘‘‘‘‘‘‘‘ —

._ g/ / B ' V \ ~ 4444 E __.__________.__.;_8 _

-—-4 _ ————— —_v —
02 V ’’’’’ ’V— ,A

E 0.45 T v— —————— v- ~~~~~~ V ,,,,, / /- _

0O _ / ’. —————————— 0’ _

I o. ”/t’ / -

_ \‘/,, _

0.40 _ _
0.35 l I I I I I L I 1 I l I I I I I

12.5 14.0 15.5 17.0 18.5 20.0

x (cm)
0 Unforced case, D A = 2°, V A = 4°, 0 A = 8°

Figure 28. Streamwise evolution of the mixed fluid fraction 8,, /51 as a function of the
forcing amplitude A compared to the unforced case.

 

 

 

 

 

 

 

 

0.55 I I I I I I I I I I I I I I
- (b)f= 4 Hz _
_ --e ______ J
0.50 _ ______ 0 ____________ O “““““ _
. a; o ------------- o _
_ -‘T'VEI\ -...-EI -
\ _.——-"""
.. _ ‘B-\ -/‘ -
to _____ V“ -\‘\B/’/
9— \
Va 0.45 — 40.6, i
00 ” \\ "I
\ x9
_ \\ //7 _
I‘ ._ -------- _.\\ \\ I’V/ I,/ .-
._ \ ,I" _
\ 2" /
0.40 _ ~\. ----------- —o 5
O 35 I l l I I I 1 l I I l I I I
12.5 14.0 15.5 17.0 18.5 20.0
x (cm)
0 Unforced case, D A = 2°, V A = 4°, 0 A = 6°
0.55 I I V I I I I I I I I I I I
_ ’. _
_ (c)f— 8 Hz x/fl _
_ ,—G ______ /, L/ __
0'50 ._ ,-’V"-" “““““ “G" ............ O-___~’$<//"//q ’/’/'/B _
_ G [*7 fix """""" O _
/ /
I B‘“‘V* ~13 ’// ’/’ I
-— _ - \ / _
‘8 \~ /=r-—"
E 0.45 - Ev _
00 _ ’_____._{_v/ _
/V" ’
r— /// -I
~ V/ I/./ _
r"/,- _
0.40 V _
O 35 I l I I I I l I I I I I 1 I
12.5 14.0 15.5 17.0 18.5 20.0

x (cm)

0 Unforced case, B A = 2°, V A = 4°, 0 A = 6°

Figure 28. Streamwise evolution of the mixed fluid fraction 5,, /51 as a function of the

forcing amplitude A compared to the unforced case.

53

within the shear layer is mixed (to within the measurement resolution 200 pm).

The amount of mixed fluid in the center of the layer is characterized by the peak
of the mixed fluid probability curve (pm)m. The amplitude dependence of (pm)max is
shown as a function of x in Figure 29. It is observed that there is more mixed fluid in
the center of the unforced shear layer compared to all of the forced cases. For the cases

forced at f = 2 Hz, (pm) is in general decreasing with amplitude and downstream

distance. The 4 Hz cases exhibit a similar behavior with the exception of a slight
increase at x = 20.0 cm for the two larger amplitudes. When forced at f = 8 Hz, A = 2°

(pm)um initially decreases then continually increases; while at higher amplitudes the

amount of mixed fluid in the center of the layer is observed to continually increase. It

 

 

 

 

1.0 I I I I I I I I I I I I I fl If I
i 1
_ ——::_-_-:--_-:" :::='——==B~':'\" ------- O ------------ .3 ............. _
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09 — V“ ““““ V VVVVVV V ___________ .3 _
. _ v ““““ ~V _
_ ._ ________ ‘ :
$0.8 _ k \ , / ._ ________ _
E — \“fi zzzz T‘ --------- O _
E - _
CL - _
0.7 __ :
0.6 :‘ j
— (a)f= 2 Hz —
05 P I I l I I l I l l I 1 l I I I I ‘

12.5 14.0 15.5 17.0 18.5 20.0
x(cm)

o Unforced case, U A = 2°, V A = 4°, 0 A = 8°

Figure 29. Streamwise evolution of the amount of mixed fluid in the center of the layer
as a function of the forcing amplitude A compared to the unforced case.

 

 

 

 

 

 

 

 

 

10 I I I I I I I I I I I I I I I I
Z G ------------- 0 ------------ e ------------ <3 ------------ G ............. e Z
__ I3— ‘‘‘‘‘ —EI-\ V
._ ‘\~ ._
0.9 _ \B\ -
" W\ -\B— -------- _‘
_ \\\\ V‘E] R“VEI _
. \v\\\ _
_ “_ \\ __
$0.8 — \‘-\-~\ V\\ _,
E ._ .\ \\\ _
E I— ‘ \‘ \\V_ —
13.1 _ ‘\‘ VVVVV ~9— VVVVV ‘V _
_ \ __
0.7 _ ‘.\‘~\ ‘1
— \‘~\.‘\-_~_‘ /‘_,/"/—. d
... V\“.‘- —
0.6 r 4
: (b)f= 4 HZ :
05 h I I I I I I I l I I I I I l l I d

12.5 14.0 15.5 17.0 18.5 20.0
x (cm)
0 Unforced case, B A = 2°, V A = 4°, 0 A = 6°
10 I I I I I I I I I I I I I I I I _
: o ------------ 9 """""" 9 """"""" {3 ---------- <3 ------------- e :
0.9 f 5K V_
I- \ \ ’//‘E -
V \ l’FV/- fif—f-Vj'j'ffi ’ I I ‘
g 0.8 _— EN _____ ‘9’/:’7—-;/ —_
E _ It"/// _
E - I'C’Jv/ 4
Q1 __ /’¢"é/ 4
0.7 - ,//fi’ 4
._ // / —I
—- V’ l/’ -I
_. /' 4
_ g" 4
0.6 V V
— (c)f= 8 Hz —
05 F I I I L I I I l I I I I I l I I -
12.5 14.0 15.5 17.0 18.5 20.0
x (cm)

0 Unforced case, C! A = 2°, V A = 4°, 0 A = 6°

Figure 29. Streamwise evolution of the amount of mixed fluid in the center of the layer
as a function of the forcing amplitude A compared to the unforced case.

55

is interesting to note that (pm)mam reaches a minimum (in the 4 Hz cases) at x * z 1, when

the structures saturate; this will be further discussed in section 3.4.

3.4 Nondirnensional streamwise variation of 51, 8 (Sm/81 and (pm)

m ’ max

The previous sections discussed the evolution of the shear layer in terms of the
downstream coordinate x; now flow evolution will be discussed in terms of the
nondimensional streamwise coordinate x *= Ax f / Uc. The shear layer width 81 and
mixed fluid thickness 5m normalized by the forcing wavelength UC / f are shown in

Figure 30 and 31 versus x *. These Figures display all of the forced cases from Figures

 

 

 

 

12 I I I I I I I
_ /W_ _
9 — /’E,-.—a/e--5% —
_. ___‘e’ )I' _
s /®/,€B /+ ,—+" _
s ,‘E , 4" _
20 F 6 /+/ ‘
00 ~ _
_ gffi _
.—.’
_ of; _
3 ,_ _
O I I l I I I I _
O 0.5 1.0 1.5 2.0

*

X
of=2Hz,A=2°,Df=2Hz,A=4°,Vf=2Hz,A=8°
0f=4Hz,A=2°,If=4Hz,A=4°,Af=4Hz,A=6°
+f=8Hz,A=2°,ef=8Hz,A=4°,®f=8Hz,A=6°

Figure 30. Nondirnensional streamwise evolution of the shear layer width normalized by
the forcing wavelength for the various forcing conditions.

 

56

 

 

 

 

5 I I I I I I )8
_ ,g/JB _
/ x'
_ E _
4 h i /’+ -
~ fl ’4‘ _
_ , g / _
._ ’I” /+ _
— ’+”
If, / -‘
03 — $’/+ —
D ' _
<4} ' _
E _ _
00 r _
2 b {‘6 _
1 ” M -
0 .— I I I I I I I _
O 0.5 10 1.5 2.0

of=2Hz,A=2°,Df=2Hz,A=4°,Vf=2Hz,A=8°
0f=4Hz,A=2°,If=4Hz,A=4°,Af=4Hz,A=6°
+f=8Hz,A=2°,ef=8Hz,A=4°,EIf=8Hz,A=6°

Figure 31. Nondirnensional streamwise evolution of the mixed fluid thickness normalized
by the forcing wavelength for the various forcing conditions.

26 and 27 in nondimensional form. As expected, with increasing x’" the shear layer width
continually increases. The saturation of the structures may also be seen in the frequency
locked region, near x *= 1.5 . The amplitude effects are also observed to increase the
shear layer width. Similarly, the amount of mixed fluid increases with increasing x *.
There is, however, no saturation effect in the frequency locked region.

The mixed fluid fraction 5m / 81 is displayed in Figure 32 as a function of x *.

Noteworthy features of this Figure are a local minimum occurring near x *= 0.8; and a

 

57

 

 

 

 

0.55 I I I I I I I
_ ’g -
050 _ a” _
. " , ’ 7+ -
: (596090 g y /+/ _
Io ,

‘ \ /' +x 7' /’ -
°°fi _ a; I I .‘u/. T \ i +’ ‘
\e 0.45 — ‘~. g‘ _
°° ” ESE}; \ 9’76 ‘

_ W/ ‘. fl x / _

_ V / M ‘I .1/ Q j ...

_ g \ I- g/ _

0.40 ”‘ k4 -
O35 _ I I I I I I I _
0 0.5 1.0 1.5 2.0
X*

Of=2Hz,A=2°,Df=2Hz,A=4°,Vf=2Hz,A=8°
0f=4Hz,A=2°,If=4Hz,A=4°,Af=4Hz,A=6°
+f=8Hz,A=2°,ef=8Hz,A=4°,®f=8Hz,A=6°

Figure 32. Nondirnensional streamwise evolution of the mixed fluid fraction for the
various forcing conditions.

general increase in the mixed fluid fraction for x * > 1.0. Figure 33 shows the probability
of mixed fluid in the center of the layer (pm)max, which is proportional to the amount of
mixed fluid in the central of the layer. It is seen that (pm )m, in general, decrease while x *
< 1.0 and increase for x "’ > 1.0. These observations are connected to the growth rate,
size and spacing of the structures. The local minimum appears to be connected to the
enhanced growth rate of the layer; in this region the layer width is increasing faster than
the amount of mixed fluid in the layer. In the frequency locked region the opposite is

true, layer growth rate is not increasing as fast as the amount of

58

 

 

 

 

1.0 _ l I l I I I I _
: GOQOGO F0 :
0.9 : I“ Ii .—
: 'x "N \ ,5; :
_ l\ \ 7E7 _
x0.8 — W I xx ”x _
('5 ‘ \y \+/ -
E _ V vv )2 _
E ' I I S ‘
a. _ \ I / _
0.7 f x E j
_ \A /3 691/ _
_ I! g -
0.6 : j
05 H I I I I I I I q
0 0.5 1.0 1.5 2.0

X*

of=2Hz,A=2°,I:If=2Hz,A=4°,Vf=2Hz,A=8°
of=4Hz,A=2°,If=4Hz,A=4°,Af=4Hz,A=6°
+f=8Hz,A=2°,$f=8Hz,A=4°,lZf=8Hz,A=6°

Figure 33. Nondirnensional streamwise evolution of the amount of mixed fluid in the
center of the layer for the various forcing conditions.

mixed fluid in the layer, see Figures 26 and 27. Physically, the reason that the frequency
locked region, ie. 8 Hz cases, has more mixed fluid is that the structures, although
roughly the same size as the 4 Hz cases, are twice as many in number.

The above nondimensional analysis used the shear layer width 81 and the forcing
wavelength UC / f as length scales. The shear layer width, recall is computed similar to
boundary layer thickness, in that the edge of the layer is defined by the points where the
total mixed fluid probability curve reaches 1% of its maximum value. It suffers from

resolution problems, due the non-reacting experiments used in this work over-estimating

 

59

the amount of mixed fluid. In some cases the pm( y) curves asymptotically approach
zero, artificially increasing the shear layer width. Perhaps a more robust length scale
would be the concentration thickness 05, an integral quantity analogous to momentum
thickness (introduced in section 2.4) based on average concentration measurements. The
relationship between the concentration thickness and momentum thickness is displayed
in Figure 34. For the unforced layer the relationship is nearly linear, as the frequency and
amplitude increase there is an apparent scatter in the plot at large 81. This scatter may
be the result of deficiencies on the part of both 81 and 05. The concentration thickness

may suffer when there is a well developed flat region on the average concentration

 

 

 

 

profile.
6 F I I I I I I I I I I I I I I I I I I
- (a)f= 2 Hz —
_ C ..
_ b _
5 _ ,/” _
l"’
I— i' —l
M h "V. 5
® 4 ’— .r”/ /v —.
F— ' ”O i
- /,.0’
;€;;' _
' -4343
3 - (53/ —
2 I I I I I I J L L I I I I I I I I I I
20 25 3O 35 4O
5

O Unforced case, C! A = 2°, V A = 4°, 0 A = 8°

Figure 34. Streamwise evolution of the concentration thickness as a function of the
forcing amplitude A compared to the unforced case.

 

 

 

 

6 I I I I I I I f I I I I I I I ’l I I I
_ x. .u
b— // —4
_ (b) f_ 4 Hz /, _
_ /.y _
I— ,’ ._

5 / W/
.— ,. / ..
,/ /
i— ’/ / d
V I/’ / _
5U" " )Ififfl/ ‘
® 4 _ //3’ [Z ""
‘ ///” 42f ‘
1’ / ',-O
F /W 410" d
/ "

._ / 5' -
_ / ”g—g’e _
3 - 013/ -

2 I I I I I I I I I I I I I I I I I I I

20 25 30 35 4O

 

 

 

 

I I I I I I I I I I I I I I I I
I I I
_ ._. ._
— (c)f= 8 Hz ,/ —
_ /’H _
._ ,’/' N _
5 ,r ,AV/
_ ,/ / _
’/'///V’V d
,,{a/’ -251
_ / / _.
® 4 _ //’B’/. _
_ v’,/ _
I— ’/ "’0 .—
,Z x -o"

_ ({Ii‘0””0 _
3 — o'Ij/ _.

2 I I I I I I I I I I L L I I I I I I I

20 25 3O 35 40

o Unforced case, D A = 2°, V A = 4°, 0 A = 6°

Figure 34. Streamwise evolution of the concentration thickness as a function of the
forcing amplitude A compared to the unforced case.

 

61

The concentration thickness normalized by the forcing wavelength 66 f/ U6 is

shown versus x " in Figure 35; note the similarity to Figure 30.

 

 

 

 

16 I I I I j I I I I I I I I I I I I I I
_ ’B—fll/ _I
— fl/ ,-e--"EB _

.e-"V'
1.2 *— Rf ,9” /,+ _
I_. ’1’ ~‘_I(’ _
$, /+
o ,+/
E b / "
“H 08 — 4.x 4/ ~
M .
G h f}! A
.-
_ ,I _
.l ,r'
“ l’ ,0 _
’./.

0.4 — I _
_ gvvg -

0 I I I I I I I I I I I I I I I 4 I I I
0 O 5 1.0 15 2O

of=2Hz,A=2°,Df=2Hz,A=4°,Vf=2Hz,A=8°
.f=4HZ,A=2°,-f-_-4Hz,A=4°,Af=4Hz,A=6°
+f=8Hz,A=2°,ef=8Hz,A=4°,I21f=8Hz,A=6°

Figure 35. Nondirnensional streamwise evolution of the concentration thickness
normalized by the forcing wavelength for the various forcing conditions.

The mixed fluid thickness normalized by the concentration thickness 5m MFR Figure 36,
shows a trend very similar to the mixed fluid fraction versus x *, see Figure 32. The
most notable feature, again, is the local minimum occurring in the vicinity of x * = 1.0,

the end of the enhanced growth rate region. As x * increases past 1.0, a steady increase

62

in 8m NilE is observed. Trends in Figure 36 are the same as those shown in Figure 32, the

only difference being the lack of the scatter in Figure 32.

 

 

 

 

4.0 I I r T I I I I I I I I I I I I I I I
: 000 :
_ ’06 t. _
3 5 — 0 I256 “v —
. _ [Fifi t. I e
_ l \\ $.57 IZI :
_ i 5i: é’l” ,/ _
w" l \ / -fi
® r I I, x -
\E 30 __ W Z\ ‘I F _
°° ’ Al I e/ ‘
I
_ \ 69"“63’ I21 _
_ / s
_ K A E, -
2.5 — ,1 w _
2.0 I I I I l I I I I I I I I I | I I I I
0 0 5 1.0 1 5 2 O

Of:2HZ,A:2°,CIf:2HZ,A-‘=4O,Vf=2HZ,A-—-8o
Of:4HZ,A:2°,If=4HZ,A=4o,Af'—'4HZ,A=6O
+f=8Hz,A=2°,ef=8Hz,A=4°,®f=8Hz,A=6°

Figure 36. Nondirnensional streamwise evolution of the mixed fluid thickness normalized
by the concentration thickness for the various forcing conditions.

The plots in this section were normalized with respect to the forcing frequency
only; effects of the forcing amplitude are not effected by this normalization. It is not

clear at this time if the amplitude effect can be properly normalized.

3.5 Probability density function of E

63

Results thus far have addressed mixing in terms of mixed fluid at all
concentrations. The influence of forcing on the actual distribution of the mixed fluid
composition will now be discussed in terms of the average probability density function
P (é, y). According to Koochesfahani and MacKinnon (1991), the use of a single pdf to
characterize the composition field of both the unforced and the forced layers is warranted
here, since the shape of the pdf was observed to be nearly invariant across the layer
width. The composition distribution for the unforced and forced shear layers are

presented in Appendix B.

3.5.1 Streamwise variation of the total pdf

Figures 37 displays the streamwise evolution of the total pdf of g for the unforced
shear layer. The general shape of the pdf is preserved as downstream distance increases.
An increase in mixed fluid with increasing downstream distance is seen by the increased
height of the peak of the pdf located near g z 0.75. This peak in the pdf will be referred
to as the most probable concentration. It is interesting to note how the pdf fills in around
the most probable concentration, but not at very low concentrations. It is also observed
that the peak tends to shift slightly toward lower values as x increases. Initially the layer
has an excess of high—speed fluid (E, = 1) resulting in a most probable concentration at
very high values of g. As the mixing process continues the most probable concentration
shifts toward the low—speed side (Q = 0). This result was first reported by Koochesfahani
and Dimotakis (1986).

The streamwise dependence of the pdf for the shear layer forced at f = 2, 4 and

8 Hz, A = 4° are displayed in Figures 38 — 40, respectively. The remaining forced cases

 

64

are presented in Appendix C. In all cases mixing is enhanced mostly at high values of
concentration with increasing downstream distance. Unlike the unforced layer which
shows a slight shift in the most probable concentration to lower values, some of the more

highly forced cases exhibit a slight shift toward higher values, see Figures 39 and 40.

 

 

 

 

 

 

2.5 I I I I I I I
2.0 _ _
1.5 _ _
A A
’3 - £21 . I
e 21’ "I".
1.0 — ”03.1%. X _
, B’Ox “o-&\\
I ’ ,1 \“o -
I 15.41%
0.5 F X \\ ,o I. ”0’ A
[$3 \ ’ " g0
_ \5. _ "g5 1 -
O I I I l I I I
0 0.25 0.50 0.75 1.00

O x=12.5, D x: 14.0, V x=15.5, O x: 17.0, I x=18.5, A x= 20.0 cm

Figure 37. Streamwise evolution of the total pdf of the unforced shear layer.

65
2.5 I I I I I I

 

2.0 V V

1.5 V

P(§)

1.0 V

 

 

I

 

 

 

 

0 0.25 0.50 0.75 1.00

O x* = 0.28, CI x* = 0.31, V x* = 0.34, O x* = 0.37, I x* = 0.41, A x* = 0.44

Figure 38. Nondirnensional streamwise evolution of the total pdf for the shear layer
forced at f = 2 Hz, A = 4°.

 

2.5 I I I I I

—-I
_.

2.0 V V

 

 

 

 

 

 

A 1.5 V g l A

g _ ’fl'i‘x \ _

CL , ”WES: )\

1’ J3" \

1.0 — , ,IZI’ _

z ,O’fl— 0 Kim

0.5 V V

O I I I l 4 I I
0 0.25 0.50 0.75 1.00

é
o x’“ = 0.56, a x* = 0.62, v x“ = 0.68, . x“ = 0.75, - x* = 0.82, A x* = 0.88

Figure 39. Nondirnensional streamwise evolution of the total pdf for the shear layer
forced atf= 4 Hz, A = 4°.

66

 

2.5

2.0 V

1.5 V

P02)

 

1.0—”3

l

0.5 V

 

 

 

 

O 0.25 0.50 0.75 1.00

o x’ =1.11,D xi: =1.24,V x" =1.38,0 x* =1.51, I x* =1.64, A x* =1.78
Figure 40. Nondirnensional streamwise evolution of the total pdf for the shear layer

forced at f = 8 Hz, A = 4°.

3.5.2 Amplitude dependence of the total pdf

The forcing amplitude dependence is shown in Figures 41 — 43 for the cases forced
at f = 2, 4 and 8 Hz at the farthest downstream location x = 20.0 cm, corresponding to
x* = 0.44, 0.88, 1.78, respectively. The remaining cases are presented in Appendix D.
The main result is that in all cases mixing is enhanced mostly at high values of
concentration. The effect of the forcing amplitude is simply to increase the proportion
of fluid mixed at higher values of concentration. In addition to enhanced mixing, there
is a change in the composition field. In all cases the most probable concentration is
shifted slightly toward higher values of {3. A shift in the predominate mixed-fluid

concentration was also noted by Koochesfahani and MacKinnon (1991).

67

 

 

 

 

 

 

 

2.5 I I l I I I I
2.0 V V
.. ’r’.\ _
1.5 _ I/fiw \. _
Q" _ ég-Q:E\ \ _
1.0 _ // \\\g; _
0.5 V V

0 l l L I I l I

0 0.25 0.50 0.75 1.00

0 Unforced case, B A = 2°, V A = 4°, 0 A = 8°

Figure 41. Amplitude dependence of the total pdf for the shear layer forced at f = 2 Hz,
at x = 20.0 cm (x‘ = 0.44), compared to the unforced case.

 

 

 

 

 

 

2.5 , I I I I I I I
|
_ I _
2.0 V | _
l
_ ‘ A ~
/ .
1.5 - , fl“. -
A 'QVEJ \o
“1‘ - 1’ ‘Q\ \ _
E: g x \
’87 “e‘ k.
1.0 V ,/ \\ \$ _
fi/ var
.. , ”j ‘G‘:' _
\ a a
05 —- W . ’2’? —
. \‘€\ .3555???
— gait-33229255” —
0 I l 1 l I I l
O 0.25 0.50 0.75 1.00
E

o Unforced case, B A 2 2°, V A : 4°, 0 A = 6°

Figure 42. Amplitude dependence of the total pdf for the shear layer forced at f = 4 Hz,
at x = 20.0 cm (x* = 0.88), compared to the unforced case.

68

 

 

 

 

 

2.5 I I l l I I I
c /,/ Y _
l 4 .530.
A 1‘5 — f: ‘\\v\ _
‘3? - 45' Q‘.\ \‘(I ‘
~ \
1.0 - 43,"?- ngg —
_ :\\ ’XJZ’ \‘G i _
I \ /'%€’Z ‘6)
0.5 V 0 EN , 25: V
- {ragtg‘g‘g’ _
0 1 l I l l I I
O 0.25 0.50 0.75 1.00

o Unforced case, D A = 2°, V A = 4°, 0 A = 6°

Figure 43. Amplitude dependence of the total pdf for the shear layer forced at f = 8 Hz,
at x = 20.0 cm (x* = 1.78), compared to the unforced case.

3.6 Comparison with Koochesfahani and M°°Kinnon (1991)

The current flow conditions, Ul = 40 cm/sec and U2 = 20 cm/sec, were chosen to
be the same as those in Koochesfahani and M°°Kinnon (1991), to allow comparison of the
forcing mechanism on the development of the forced shear layer. The facility used in the
current experiment was the same facility used by Koochesfahani and M°°Kinnon (1991).
In the previous work, data were presented at a single downstream position, x = 20 cm,
where the local Reynolds number R651: AU 81/v, was approximately 6,600. In the
current work Reslz 6,000 at x = 20.0 cm. Due to the finite size (height and length) of

the test section it was not possible to match the Reynolds numbers of the two

69

experiments.

As stated earlier, the unforced layer in the current study is not the same as the
natural layer, due to the presence of the airfoil; the main difference being a reduction in
the width of the unforced layer. A summary of shear layer width 51 , mixed fluid
thickness on, and the mixed fluid fraction (Sm/51, for the natural and unforced layers are

given in Table 2.

Table 2. Summary of shear layer width, mixed fluid thicknesses and
mixed fluid fraction for the natural and unforced layers.

 

 

5m (mm) 51 (mm) 8m/51
Natural 16.1 33.3 0.483
Unforced 14.4 29.6 0.486

 

The natural layer of Koochesfahani and M°°Kinnon has width 51W: 33.3 mm and the
unforced layer in the current investigation was 51 = 29.6 cm. This is approximately an
11% reduction. The reduction in the width of the layer due to the presence of an airfoil
is consistent with the results reported by Koochesfahani and Dimotakis (1989). The peak
of the mixed fluid probability curve, (pm)m, in the central region of the shear layer is
nearly the same, implying that the amount of mixed fluid at x = 20.0 cm is not changed
due to the presence of the airfoil. Therefore, the changes in 5m between the unforced and
natural layers are due purely to the reduced width of the shear layer.

In the forced layers of Koochesfahani and M°°Kinnon (1991), perturbations were

induced by sinusoidally oscillating the high-speed freestream at frequencies of 4 and 8

70

Hz, with rms amplitude of about 1.0% of the freestream speed. There is some ambiguity
in the comparing these cases to the present work since the perturbation methods are
different and the amplitudes are characterized differently. It is not clear if the methods
by which each of these forcing techniques are quantified are related to each other. The
effect of the different forcing mechanisms on the scalar mixing field does, however, show

the same trend. In Figure 44, the forcing frequency and amplitude dependence of the

f (Hz)
0 2 4 6 8
0.60 I I
._ Koochesfahani & MacKinnon (1991): -
Natural case V
V I Forced cases —

 

_I
—I

 

Present work
0-55 ------- Unforced case 4
A = 2 deg.
A = 4 deg. O
A = 6 deg. E
A = 8 deg.

V

OOD<I

/51

e 0.50 ~

............................................................................................................

 

0.45 — I _

 

 

 

040 l I l I I I I

Figure 44. Nondirnensional streamwise (frequency) dependence of the mixed fluid
fraction for various forced cases compared with Koochesfahani and
M°°Kinnon (1991).

71

mixed-fluid fraction for the two forcing mechanisms are displayed. At a fixed
downstream location the nondimensional streamwise coordinate x * = A x f / Uc is directly
proportional to the forcing frequency f. Therefore the frequency scale of Figure 44 may
also be viewed as an evolutionary scale. Only for the cases forced at f = 8 Hz is there
more mixed-fluid per unit width of the layer than in the unforced case at this downstream
location; if the flow is examined farther downstream, further increases in the mixing
would be expected. Table 3 summarizes the shear layer width 51 , mixed fluid thickness
5m and the mixed fluid fraction 5m/81 for the forced cases in both of these studies.

Table 3. Summary of shear layer width, mixed fluid thickness and mixed
fluid fraction for the different forcing conditions.

 

 

5m (mm) 51 (mm) 561/51

Koochesfahani and M°°Kinnon
f: 4 Hz 18.3 40.5 0.452
f: 8 Hz 19.3 37.5 0.515

Present work

f: 2 Hz, A = 2° 14.2 29.4 0.482
f: 2 Hz, A = 4° 14.4 31.0 0.463
f: 2 Hz, A = 8° 15.7 34.7 0.453
f: 4 Hz, A = 2° 14.5 30.6 0.472
f: 4 Hz, A = 4° 14.8 34.2 0.433
f: 4 Hz, A = 6° 15.6 36.0 0.434
f: 8 Hz, A = 2° 15.5 31.2 0.498
f: 8 Hz, A = 4° 17.6 34.0 0.517

f: 8 Hz, A = 6° 18.5 35.2 0.526

 

72

The total pdf for the oscillating freestream experiments showed trends similar to
the present work. In both cases the effect of forcing was to increase mixing at mainly
higher values of concentration. It is noted, however, that the predominant concentration
of mixed fluid shows only slight variations relative to the unforced case, in contrast to
the oscillating freestream experiment which showed a larger shift to higher concentrations.
It is not clear at this time why the current forcing method results in smaller shifts of the
predominant mixed fluid concentration. It is possible that the wake of the airfoil may be

a contributing factor.

3.7 Comparison with Roberts and Roshko (1985)

In the work of Roberts and Roshko (1985) the effect of a periodic disturbance on
the amount of chemical product in a forced shear layer was examined at pre-transitional
and post-transitional Reynolds numbers. Forcing was achieved by sinusoidally oscillating
the high-speed freestream, at 2% (rms) of the freestream speed. The chemical product
was measured using a laser absorption technique. In this work the amount of mixing was

quantified by the product thickness, defined by

+h

1 _
5p, 2 EV [9,0061%

20 -h

where C2 is the low-speed feestream reactant concentration and 6'; (y) is the average
0

product concentration defined as

1—8

E;0)=Czo f (1-€)p(€,y)d5.

73

The product thickness for the various forcing conditions was noted relative to the product
thickness of the unforced layer, denoted by the ratio (8p2)F/(5p2)N’ where 81,2 is the
product thickness and the subscripts F and N refer to the forced and natural layers,
respectively.

In these experiments the unit Reynolds number Re / = AU / v of the pre-transitional
(low Reynolds number) case was 905 cm"/sec, and for the post-transitional (high
Reynolds number) case 4340 cm‘I/sec. The natural frequency for the low Reynolds
number case was f0 = 8.3 Hz and for the high Reynolds number case f0 = 62.7 Hz. In the
current work the unit Reynolds number is 2000 cm‘I/sec, corresponding to the later stages
of the mixing transition. The natural frequency for the unforced case is f0 z 27 Hz.

It was found, in the work of Roberts and Roshko, for the low Reynolds number
case that forcing had steadily increased the ratio of the product thicknesses (8122) F/ (8P2)N
through the enhanced growth region, reaching a local maximum near x *= 2. Once past
the frequency locked region the product thicknesses steadily decreased. The ratio of
forced product thickness to the natural product thickness (8102) F/ (5P2) N is shown in Figure
45 as a function of x * for the low Reynolds number case. In the high Reynolds number
case, the forced product thickness is initially as much as 4 times larger than the unforced
product thickness; their ratio (5P2) F/ (81,2) N steadily decreases through the enhanced growth
region. In the frequency locked region and beyond, x *> 1, the ratio of the forced to the
natural product thickness remains slightly less than the one, see Figure 46. This indicates
that there is actually slightly less chemical product, on average, in the forced layer than
in the unforced layer.

In the present work, the ratio of the forced mixed fluid thickness to the unforced

 

 

 

 

 

* . .D F-1.BHz. Aboorption
O F-2.8Hz. Abeorption
g. A F-4.6flz. Absorption
* F-8.8Hz. Absorption
~ 0 F-5.0 Hz. Fluoreecence.
fl V F-2.6 Hz. Fluoreecenoe
0 F- 1.0 Hz. Fluoreeoence
b * *
* i
i
A e
A .
A A
II] A A o A A 0 A *
Rib 00° 0 A0 at
F-e- - --- ------------------------------------ —
o 0%!)
0 1.0 2.0 3.0 4.0 6.0
1x!
U

Figure 45. Ratio of forced product thickness to natural product thickness as a function
of x’ : low Reynolds number case, reprinted from Roberts (1985).

4.0

3 . O
(8pZ)F
(8p2)N

2 . 0

10°

 

 

 

 

D ”more. Moot-pupa
§ F-0.0Hz. Aoeorpuon
F-O Hz. Fluoreooence
F-ta Hz. Fluoneeoenoe
I- 0 F-aom. Fluoreeoenoe .
O
[J .
O
D
O
I- a -
00
O
03
”glam-"o ------------------------------------- —
0000000 A °
0 1.0 2.0 3.0 4.0
le
U

Figure 46. Ratio of forced product thickness to natural product thickness as a function
of x’" : high Reynolds number case, reprinted from Roberts (1985).

75

mixed fluid thickness (8m)F/(8m)U is analogous to the ratio of the forced to natural
product thickness, see Figure 47. It should be noted that the current experiments were
done using a non-reacting technique which over-estimates the amount of mixed fluid.
Therefore, the trends shown in the data are of interest not the actual values of the
quantities. The trends observed in the present data are most similar to the trends shown
in the low Reynolds number case, Figure 45. In the enhanced growth region there is a
slight increase in mixed fluid thickness over the unforced case; and in the frequency
locked region there is a large increase in 8m that seems to peak near x *= 2. Note the

expanded scale of Figure 47.

 

 

 

 

13 _ I I I I I I I I I I I I I I I 18:-gl I
_ ,/ g
>— IB —
V / 9 _
1.2 — F 9 _
._ I/ _
.3 Z ,/ ° Z
E
‘3 h V W / g _
1.1 1 — V I7 A [)8 .t + _
OOE : / ,/ ,6 7p /’ :
v _ (AVA—M 2(1',’ _
V $3 .\‘..I\ I, . I +\ / ._
1.0 — [69 Odin I’Ve'i' ' _ 4' / ‘I _
0 9 7 I I I I C.) I I I I I I I 1 I I I I I I I I
0 O 5 1.0 1 5 2 0
X*

of=2Hz,A=2°,Df=2Hz,A=4°,Vf=2Hz,A=8°
ef=4Hz,A=2°,If=4Hz,A=4°,Af=4Hz,A=6°
+f=8Hz,A=2°,€Bf=8Hz,A=4°,®f=8Hz,A=6°

Figure 47. Ratio of forced product thickness to unforced product thickness as a function
of x*.

Chapter 4

CONCLUSIONS

The structure and the scalar mixing field in a shear layer forced by an oscillating
airfoil were investigated over a range of amplitudes, frequencies and streamwise locations
using laser induced fluorescence in the passive scalar mode. One purpose of this study
was to compare the mixing characteristics of a shear layer which is forced by two
different methods. Comparisons were made with previous results from a shear layer
perturbed by the oscillation of one freestream.

It was observed that. the general structure and the growth characteristics of the
forced shear layer here were similar to previous results. As either the forcing frequency
or amplitude was increased, there was, in most cases, a significant increase in the shear
layer width. Results show a reduction of the amount of mixed fluid in the central portion
of the forced layer, an increase in the width of the region where mixed fluid is found, and
an increase in the total amount of mixed fluid in the layer. The fraction of the layer
width occupied by mixed fluid was found to vary slightly, however, when compared to
the unforced case. These results are consistent with those from a shear layer forced by
the oscillating freestream method. The predominant mixed fluid concentration showed
similar variation as a result of forcing, when compared to previous results from oscillating

freestream forcing studies. The composition distribution for the unforced and the forced

76

77

shear layers was reported to be essentially uniform across the width of the layer again.
This result is consistent with the oscillating freestream experiment.

The effect of downstream evolution was found to increase shear layer width and
increase the total amount of mixed fluid in the layer. The amount of mixed fluid in the
center of the layer and the mixed fluid thickness were found to vary depending the stage
of the evolution as defined by the nondimensional streamwise parameter x “ = Ax f / UC.
A local minimum was reported near the end of the enhanced growth region for both the
mixed fluid fraction and the amount of mixed fluid in the center of the layer. The choice
of length scale when computing the mixed fluid thickness as noted to have a slight effect
on the reported efficiency of the mixing layer. In most cases the unforced layer was
observed to be more efficient, in terms of the amount of mixed fluid per unit width of the

layer, than the forced layer.

APPENDICES

APPENDIX A

Amplitude dependence of the transverse profiles

Amplitude dependence of the transverse profiles of average concentration E, total
mixed fluid probability pm, pure low-speed fluid probability p0 and pure high-speed fluid
probability p1 at the downstream locations not presented in Section 3.2.2 are shown in

Figures A.1 - A15.

78

p0

1.0

0.8

 

 

l I I I I I I I I I

 

 

 

 

 

 

 

 

79

pI

 

 

 

 

 

 

 

  

 

 

1.0 I I I I | I I I I l I I I I I I I r r
0.8 V —
0.6 V A
0.4 V —
0.2 V A
0 gig I I I 1 I I 1 L1 1'“
-20 -10 O 10 20
y(mm)
(b) Total mixed. fluid probability
1-0 ' r I ' I ' ' I ' I I I fi—F—
0.8 V ' -
0.6 V —
0.4 — —
0.2 V ' -
.I‘II’
O I I I I l I I I I 1
-20 -10 0 10 20
MM)

((1) Pure high—speed fluid probability

0 Unforced case, C] A = 2°, V A = 4°, 0 A = 8°

-20 -10 0 10 20
y (m)
(a) Average concentration
1.0 m! f I I I l I I I I l I I I I j
_ R J
0.8 V V
0.6 V V
0.4 V V
0.2 _ 5‘ _
_ 'I
O I I I I I I I I I V
-20 -10 0 10 20
y (mun)
(c) Pure low-speed fluid probability
Figure A.1.

Transverse profiles for the shear layer forced at f = 2 Hz as a function of

the forcing amplitude A, at x' = 0.28, compared to the unforced case.

80

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

I.0"‘TfT"‘l"' 1.0 T'ITII"'I'I"I"'I
_ .4 V? _
V 0.8 _ «q e
I
I», -
I l I I I I I I I I l I I IL
10 O 10 20
y (m) y (m)
(a) Average concentration (b) Total mixed fluid probability
1.0 I I I I I I I I I I I I I I I 1.0 I I I I I I I I I I I I I I
t Ix 3% I _ g I
'I . ’.
. ‘ ~ V 0.8 _ $6 V
I- f] _
— 0.6 ~ ’ —
c5: - _
— 0.4 _ I __
_ O 2 I— —
m5
_ g 4
d
I O I I I I I I I I I
10 20 —20 —10 0 10 20
y (mm
(c) Pure low-speed fluid probability (d) Pure high—speed fluid probability

0 Unforced case, III A = 2°, V A = 4°, 0 A = 8°

Figure A.2. Transverse profiles for the shear layer forced at f = 2 Hz as a function of
the forcing amplitude A, at x' = 0.31, compared to the unforced case.

 

81

 

 

 

 

 

 

 

 

 

 

 

  

 

 

 

 

 

 

 

 

(a) Average concentration (b) Total mixed fluid probability
1.0 F I I I I I I I I I I I I I I I 1.0 I I I I I I I I I I I I I I
5‘ $0 A _ g? _
0.8 ~ I III? - 0.8 — [I -
0.6 V V 0.6 _ _
O .—4
Q ~ cm - -
0.4 V ~ 0.4 — ”i 2
_ _ _ d . _
0.2 0.2 47%
_ - H II I
O l L I I I I O 4 m I I l I I I I
-20 10 20 -20 ~10 O 10 20
3’ (min) y (nun)
(c) Pure low-speed fluid probability (d) Pure high-speed fluid probability

0 Unforced case, B A = 2°, V A = 4°, 0 A = 8°

Figure A.3. Transverse profiles for the shear layer forced at f = 2 Hz as a function of
the forcing amplitude A, at x’" = 0.34, compared to the unforced case.

 

 

 

 

 

 

 

 

  

 

 

82

p1

 

 

-2O

-10

I I I I I l I I I I I I I I I I

   

p,
,f IE

~ ,’ éfi
0 1

IIIIIIIIII

 

0 10

y (m)

(b) Total mixed fluid probability

 

1.0

0.8 V

0.6 V

 

I I I l I I I

 

 

 

 

20

(d) Pure high-speed fluid probability

0 Unforced case, D A = 2°, V A = 4°, 0 A = 8°

1.01IIIIITIIVII'I-
0.8 V V
0.6 V V
IM - _
0.4 V ‘1 V
I- '1,
0.2 V ,’ .~ V
.— . -4
.U
0 I I l I I I I I I I I I I I I
-20 -10 O 10 20
MM)
(a) Average concentration
1.0 I I . I I I I I I
_ I! ..
u
0.8 " O @b T
‘- I
_ § \‘ _
0.6 - ‘Q —
o “a
0.. ~ g -
.\
0.4 V 1 V
0.2 V V
0 I I I I I I I I I ‘
-2O —10 O 10 20
MM)
(c) Pure low-speed fluid probability
Figure A.4.

Transverse profiles for the shear layer forced at f = 2 Hz as a function of

the forcing amplitude A, at x” = 0.37, compared to the unforced case.

83

 

 

ITII III]
1.0 I

1.0 T‘fil I I I I I I I I I I I I I I I

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

2O -20 -10 O 10 20
y (m)
(a) Average concentration (b) Total mixed fluid probability
1.0 IIVIIIIIIIIIIII 1.0IIIIIIIIIIIIIII
f! E _ _ l? _
. ‘\ - 0.8 — " -
- 0.6 ~ ' —
cf ” I
V 0.4 V V
- 0.2 e 4
' O ~’.r.--:: 14 I I I I I I I
10 20 ~20 -10 O 10 20
y (m)
(c) Pure low-speed fluid probability ((1) Pure high-speed fluid probability

0 Unforced case, CI A = 2°, V A 2 4°, 0 A = 8°

Figure A.5. Transverse profiles for the shear layer forced at f = 2 Hz as a function of
the forcing amplitude A, at x" = 0.41, compared to the unforced case.

 

 

 

 

 

1.0 IIIIIIIIIIIIT'V'
0.8V
0.6V
H—U‘ _
0.4V
_ .v‘
0.2- g
C
O IIIIIIIIIIIIII

 

 
  

 

 

 

 

84

p1

 

  

 

 

 

 

   
  

 

 

 

1.0 I I I I I I I I I I I I r l I I I I I
0.8 V —
0.6 V _
0.4 V ~
0.2 — IR“ _
- , III. ,

d {fl
0 I I I I I I I I I I
-20 —10 0 10 20
yawn)

(b) Total mixed fluid probability
1.0 I I I I l I I I I I I I IV
0.8 — Ill —
0.6 V I ~
L _
0.4 — —
0.2 _ avg _
_ .flcg _

0 w I I I I I I I I l I
-20 -10 0 10 20

MM)

(d) Pure high-speed fluid probability

0 Unforced case, B A = 2°, V A = 4°, 0 A = 6°

-20 -10 0 10 20
y (nun)
(a) Average concentration
1.0 W I T I I I I I T I I I I I I I
_ \‘% _
0.8 - III I
0.6 — fi —
o \ II
V II
0.4 V V
0.2 V V
0 I I I I I I I I I V I
-20 -10 O 10 20
y (mm
(c) Pure low-speed fluid probability
Figure A.6.

Transverse profiles for the shear layer forced at f = 4 Hz as a function of

the forcing amplitude A, at x" = 0.56, compared to the unforced case.

 

85

 

 

 

 

 

 

 

 

 

 

1.0 WI:
0.8 V
0.6 V
HU‘
0.4 V
,. /‘/I
()2 — !/,
4 a
I ,1 Va
,0
0 I_L I I I I I I I I I
-20 -10 O 10
NM) y(mm)
(a) Average concentration (b) Total mixed fluid probability

 

 

 

 

 

 

 

 

 

 

1.0 I I I I I I I I I I I I I I I I 1.0 I I I I I I I I I I I I r
.. ‘.\ WI ’ WEE] _
_ I! \' ,I ,/ _
0.8 ~ I,
M
_ \\ _
.‘\
0.6 V ‘ . ._
a? - I35 _
0.4 V V
0.2 V _
O I I I I I l I I I I -‘- < - '3 I I I I I I I I
-20 -10 O 10 10 20
y (m)
(0) Pure low-speed fluid probability ((1) Pure high-speed fluid probability

0 Unforced case, D A = 2°, V A = 4°, 0 A = 6°

Figure A.7. Transverse profiles for the shear layer forced at f = 4 Hz as a function of
the forcing amplitude A, at x’ = 0.62, compared to the unforced case.

86

 

 

1.0

0.8 V

0.6 V

0.4 V

 

 

 

 

 

O IIIIIIIIIIIIIII

-20 -10 O 10

y (m)

 

(a) Average concentration

 

 

 

 

 

 

 

 

 

 

I l I I I I I T T I I I
10
y (Hun) y (mrm
(c) Pure low-speed fluid probability (d) Pure high-speed fluid probability

0 Unforced case, B A = 2°, V A = 4°, 0 A = 6°

Figure A.8. Transverse profiles for the shear layer forced at f = 4 Hz as a function of
the forcing amplitude A, at x” = 0.68, compared to the unforced case.

 

 

 

 

 

 

 

 

 

87

 

 

 

 

 

 

 

 

 

1.0 I I
0.8 V ' V
0.6 V V
a? I -
0.4 V ‘79] V

(P
02— QID “ I
9: é‘D I x
- f ob“ _
O I l I I I 4 l I I I $—
—20 -10 O 10 20
ymm)

(b) Total mixed fluid probability
1.0IIIIIIIIIIIIIW
03— $3 I

II" -
If
0.6 _ :13, ‘
a: ' I
0.4 V _
J _
02~ ¢§y -
_ “$9, 339
0 IN I I l I J
—20 -10 O 10 20
ymm)

(d) Pure high-speed fluid probability

0 Unforced case, I: A = 2°, V A = 4°, 0 A = 6°

1.0 I T I I I
0.8 V
0.6 V V
um I '
04 V f‘vcgo —
.— I’ll? W
02— 97W -
_ ,mv
‘ )7
O I I I I I I I I I I l
-20 -10 0 10 2O
ymm)
(a) Average concentration
1.0 x I I I I I l‘r I I I I I I I
_ I Q.
0'8 ” ‘6. I39 ‘
’ I We?
06- gm, —
a? \,\
III.
0.4 V ‘q V
_ b
0.2 V V
o ‘0
O I I I
-20 -10 0 10 20
ymm)
(c) Pure low-speed fluid probability
Figure A.9.

Transverse profiles for the shear layer forced at f = 4 Hz as a function of

the forcing amplitude A, at x" = 0.75, compared to the unforced case.

 

 

 

  

 

 

 

 

 

1.0 I I I I I I I l
- g‘g‘ _
0.8 V V
0.6 V V
IIU‘ Va; _
v
0.4 — if 54¢ —
_ r ya _
0’ ,
l7 ,
0.2 V f); a? V
_ f V ,,
O 1 I I I I I l I I I I I I I I I
-20 —10 0 10 20
y(mm)
(a) Average concentration
1.0 WI I I . I I
” «IRE? l
0.8 — «If? —
r ‘I V Eb
II II
0.6 ” fl??? ‘
a? - \IIII
0“”?
0.4 - MI. V
0.2 ” I‘ ‘
0 I I I I l I I 1M
-20 -1O 0 10 20
y(mm)

(c) Pure low-speed fluid probability

88

1.0

0.8

 

 

 

 

 

 

 

 

 

I I I I I I I I I
009
5’ on ‘
I" WV Q? _‘
”.0005 ‘ _
“ R
III“? III ‘
III III I
‘I £33?“ _
I‘ If W ‘I\\WI‘

o , ‘ I
"l Wig ‘ &b
M I l I I l I 1&—

-10 O 10 20
y (m)
(b) Total mixed fluid probability
-10 O 10 20
y (Inna)

((1) Pure high-speed fluid probability

0 Unforced case, D A = 2°, V A = 4°, 0 A = 6°

Figure A.10. Transverse profiles for the shear layer forced at f = 4 Hz as a function of
the forcing amplitude A, at x’ = 0.82, compared to the unforced case.

 

89

 

 

 

 

 

 

 

 

 

 

 

1.0'IIIIIIIVIIII 1.0""l""IT"'l"l'
0.8 V V 0.8 V $ 59% V
I V . "
0.6 _ 06 '— O.‘I _
ls-U‘ _ me ya u _

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0.4 — — 04 — I I I ~
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0.2 — - 0.2 — I? (III V
/ I‘\Ib
_ I I 5; III _
O I I I l I I I I 0 l I I I I l I I I L
—20 10 20 ~20 -lO 0 10 20
MM) y(mm)

(a) Average concentration (b) Total mixed fluid probability
1.0 WI I I I I I I I I I I I I I I 1.0 I I I I I I I I I I I IfF—
_ _ _ ’ 7" -

I1 [I]
0.8 — I \I I 0.8 — I'M V
I "I/

‘ 'II II ” II; I
0.6 V § V 0.6 V W V

p0
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f
80
p1

0.4 - ”M — 0.4 -

0.2 — EX — 0.2 — ,
F u d b ‘v

 

 

 

 

 

 

 

v
O I I I | I I I I . 0 AM I I I I l I I I I
-20 -10 0 10 20 -20 -10 O 10 20
y (m) y (mm
(c) Pure low-speed fluid probability (d) Pure high—speed fluid probability

0 Unforced case, D A = 2°, V A = 4°, 0 A = 6°

Figure A.11. Transverse profiles for the shear layer forced at f = 8 Hz as a function of
the forcing amplitude A, at x'" = 1.11, compared to the unforced case.

9O

 

 

1 .0 I I I I I I I I I I I l I ‘
I
_ g]! _

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

-20 -1O 0 10 20

 

y (m)
(c) Pure low-speed fluid probability (d) Pure high-speed fluid probability

0 Unforced case, D A = 2°, v A = 4°, 0 A = 6°

Figure A.12. Transverse profiles for the shear layer forced at f = 8 Hz as a function of
the forcing amplitude A, at x’“ = 1.24, compared to the unforced case.

91

 

 

 

 

 

 

 

 

 

1.0 1.0
0.8 V 0.8 V
0.6 V 0.6 V

ILU‘ _ Gas

0.4 — 0.4 —
0.2 - , 0.2 —
- .I d I _

O .fii l I I I I l I I I I I I I I I O
-20 -10 0 10 20 -20

 

 

 

 

 

 

 

 

 

y (mm

 

(c) Pure low-speed fluid probability ((1) Pure high-speed fluid probability
0 Unforced case, D A = 2°, v A = 4°, 0 A = 6°

Figure A.13. Transverse profiles for the shear layer forced at f = 8 Hz as a function of
the forcing amplitude A, at x’ = 1.38, compared to the unforced case.

 

92

1.0 I I I I I I T I I I I I I f 1.0 ‘I I I I I I I I I I I 1‘17! I I I I I
900
_ 72¢ _ _ g Q0 _
V/I ’ \

d ¢

0.8 '- -¢‘ 9.271 7 7 I — 0-8 h I aD‘I —
— "2;: ‘ d _ if —
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0.6- ’30 - .—
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ILU‘ L ’J/VPVJ' -QE _ flgw Rigs -
Q

 

 

  

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

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0.4 — [JV/$101 _ 0.4 _ #95 «I? V
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1.0 II'I'IIT'III'I' 1.0MIIITIIIIIIIr ‘
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I I I L l I I I I I I
0
-20 -10 O 10 20
y(mm)
(c) Pure low-speed fluid probability ((1) Pure high-speed fluid probability

0 Unforced case, B A = 2°, V A = 4°, 0 A = 6°

Figure A. 14. Transverse profiles for the shear layer forced at f = 8 Hz as a function of
the forcing amplitude A, at x' = 1.51, compared to the unforced case.

 

93

 

 

1.01lllllIIIIIITV 1.0 fiIIIIIIIIIIIIIIIrIr
I
\
VJ ‘.'""-
‘7, v --
c r . ’ w
.r"" v 'x
a ‘0

 

 

 

 

 

 

 

 

 

Lowau‘gswwrwrmwww
Q, I
0.8— {’85

0.6 _ ‘\<7\‘é?

p0

0.4 L— ‘. \ fl} 2“

 

 

 

 

-20 -10 0 10

 

y (mm)
(c) Pure low-speed fluid probability (d) Pure high-speed fluid probability

0 Unforced case, D A = 2°, V A = 4°, 0 A = 6°

Figure A.15. Transverse profiles for the shear layer forced at f = 8 Hz as a function of
the forcing amplitude A, at x' = 1.64, compared to the unforced case.

APPENDIX B

Distribution of mixed fluid composition

The distribution of mixed fluid composition across the height of the test section
for the unforced and forced shear layers are shown in Figures B.1 - B.10 as a function
of the streamwise coordinate. From these Figures it is noted that the pdf is essentially
uniform across the height of the test section, similar to the results of Koochesfahani

(1984) and Koochesfahani and M°°Kinnon (1991).

94

y (mm)
20 0 30.0 40.0

10.0

40.0 00.0

y (mm)

Y (mm)
10.0 .

0.0

95

 

(a) x = 12.5 cm

 

-30

 

.00 0.25 0.50

 

 

0.75 1.00

 

(b) x = 17.0 cm

 

  

 

 

 

1.00

 

 

 

l
o.oao\/

0.00 0.25 0.50

Figure B. 1.

 

 

0.75 1.00

Composition distribution for the unforced shear layer.

Figure B.2.

y (mm)

y (mm)

20.0
I

Composition distribution for the shear layer forced at f = 2 Hz, A = 2°.

y (mm)

96

 

 

 

 

 

 

 
 

O
0'
V

(a) x = 0.28
O
O' __
('3
O
O
N W
O .
53'
O I l J
o'
0.00 0.25 0.50 0.75
O
0'
V

a:

(b) x = 0.37
0
d .—
CO
0
d
N

 

 
  

f—v

 
 

 

1.00

 

 

0.25

l
0.50

0.75

 

30.0

 

10.0

 

0.0

0.00

0.25

 

1.00

 

 

1.00

97

 

40.0

(a) x‘“ = 0.28

30.0
I

 

 

y (mm)
20.0
, J

 

 

 

 

O
‘3
o l I l
0'
0.00 0.25 0.50 0.75 1.00
O
:5
V
(b) x’ = 0.37
0
d >—
C0
E
\E/ .
>3

 

 

 

 

0.25 0.50 0.75 1.00

 

 

30.0

 

y (mm)
20 0

 

10.0

 

 

 

90.0

C

Figure B.3. Composition distribution for the shear layer forced at f = 2 Hz, A = 4°.

 

98

 

 

 

(a) x’“ = 0.28
C!
E0 °9 °'
£8 < ;
>x

10.0

 

 

 

0.00 0.25 0.50 0.75 1.00

 

(b) x‘ = 0.37

30.0

 

 

y (mm)
20 0

 

 

 

 

 

 

O l l 1
0o. 0.25 0.50 0.75 1.00
0
<5
V
3*
(C) x = 0.44
O
o'
CO
30
E o
vcv
>x

 

 

10.0

 

 

0.0

 

l 1 l
0.00 0.25 0.50 0.75 1.00

E

Figure B.4. Composition distribution for the shear layer forced at f = 2 Hz, A = 8°.

 

99

 

40.0

(a) x‘ = 0.56

30.0

 

 

y (mm)
20 0

 

10.0

 

 

 

0.25 0.50 0.75 1.00

 

40.0 00-0
'o
O

30.0

 

 

y (mm)
20 0

 

10.0

 

 

 

 

 

O 1 l 1
0O. 0.25 0.50 0.75 1.00
o 5
<5
V‘
*
(C) x = 0.88
C
d
0')
o?
E o O.
E o
VN
>5

 

 

 

 

 

l 1 l
0.25 0.50 0.75 1.00

E

Figure B.5. Composition distribution for the shear layer forced at f = 4 Hz, A = 2°.

 

100

 

40.0

(a) x‘ = 0.56

30.0

 

 

y (mm)
20 0

 

 

 

L
0.25 0.50 0.75 1.00

40.0 00.0

 

(b) x" = 0.75

30.0
T

 

 

y (mm)
20 0

 

 

 

0.00 0.25

 

(c) x' = 0.88

30.0
I

 

 

y (mm)
20 0

10.0

 

 

 

0.25 0.50 0.75 1.00

f

Figure B.6. Composition distribution for the shear layer forced at f = 4 Hz, A = 4°.

101

 

40.0

(a) x’“ = 0.56

30.0

 

 

y (mm)
20 0

 

10.0

 

 

 

 

l l I
0.25 0.50 0.75 1.00

 

40.0 00.0

(b) x” = 0.75

30.0

 

 

Y (mm)
20 0

 

 

l l
0.50 0.75 1.00

E

 

 

4.0.0 Coo

0.25
(C) x: = 0.88

30.0

 

O
O
U
0

y (mm)
20 0

 

10.0

 

0.0

 

0.00 0.25 0.50 0.75 1.00

Figure B.7. Composition distribution for the shear layer forced at f = 4 Hz, A = 6°.

 

40.0

30.0

 

y (mm)
200

 

10.0

 

 

 

l L
0.25 0.50 0.75 1.00

40.0 00.0

 

(b) x' = 1.51

30.0

 

 

y (mm)
20.0
/

10.0

 

 

l 1 L
0.25 0.50 0.75 1.00

 

40.0 (30.0
'o
O

(c) x* = 1.78

30.0

 

y (mm)
20 0

 

10.0

 

 

 

l L
0.25 0.50 0.75 1.00

5

Figure B.8. Composition distribution for the shear layer forced at f = 8 Hz, A = 2°.

103

 

40.0

(a) x' = 1.11

30.0
1

 

 

y (mm)
20 0

080°

10.0

 

0.25 0.50 0.75 1.00

40.0 (30.0
”o
O

 

(b) x‘ =1.51

0.0

3

 

y (mm)
20 0

 

 

 

 

O
53
O 1 1 1
O
0.00 0.25 0.50 0.75 1.00
o E
O
‘3'
III

(C) x = 1.78
O
O
CO

 

060 0

y (mm)
10 0 20.0

 

 

 

 

0.00 0.25 0.50 0.75 1.00

Figure B.9. Composition distribution for the shear layer forced at f = 8 Hz, A = 4°.

104

 

40.0

(a) x" =1.11

30.0

 

 

y (mm)
20 0

10.0

 

 

 

1 l l
0.25 0.50 0.75 1.00

40.0 c>00

 

(b) x‘“: 1.51

30.0

 

y (mm)
20 0

 

10.0

 

 

 

0.25 0.50 0.75 1.00

 

40.0 00.0

(c) x* = 1.78

30.0

 

y (mm)
20 O

 

10.0

 

 

 

l I l
0.25 0.50 0.75 1.00

E

Figure B.10. Composition distribution for the shear layer forced at f = 8 Hz, A = 6°.

APPENDIX C

Streamwise evolution of the total pdf

Streamwise evolution of the total pdf for the forced shear layers not presented in

Section 3.5.1 are shown in Figures C.l — C6.

105

 

106

 

 

 

 

 

 

2.5 I I T I I I r
2.0 V —
1.5 V —
A
(Cr , "j,
a": ' ”/2? a“ -
1.0 — ’3 £43113?) -
_ [,0"G“G\R‘j\~\ é
\‘~§ ‘
0.5 V \ -
O I I
O 0.75 1.00

 

o x’ = 0.28, D x* = 0.31, V x’" = 0.34, O x’“ = 0.37, I x" = 0.41, A x* = 0.44

Figure C.1. Streamwise evolution of the total pdf for the shear layer forced at f = 2

 

 

 

Hz, A = 2°.
2.5 ‘ l ' l ' l ‘
2.0 — r
1.5 — fl -
A I ’ \X
Lg — ’}’V-—R \\ ‘
a“ , :Et’EL\V ‘\
_ 73 ,o-- \ x
1.0 , 3’ R \ ~
I ’ ' I“
1’ I’O’

0.5 _

 

 

 

 

0 0.25 0.50 0.75 1.00

C
o x“ = 0.28, a x" = 0.31, v x’“ = 0.34, . x” = 0.37, - x” = 0.41, A x* = 0.44

Figure C.2. Streamwise evolution of the total pdf for the shear layer forced at f = 2
Hz, A = 8°.

 

107

 

 

 

 
 

 

 

 

2.5 l I l I T I I
2.0 V V
1.5 V _.
D-I
1.0 I —
0.5 — \\ -
0K; 1 —
— fieEisfit: ,- - ~
0 I l I l I l I
0 0.25 0.50 0.75 1.00
E

O x* = 0.56, D x* = 0.62, V x* = 0.68, 0 x* = 0.75, I xm = 0.82, A x* = 0.88

Figure C.3. Streamwise evolution of the total pdf for the shear layer forced at f = 4

 

 

 

 

 

 

Hz, A = 2°.
2.5 I I l ] l I I
2.0 V _
I. _
fi/‘k
1.5 — I / .9) —
53‘ _ '. 1')" ‘\'.\
E: " }/;_.E]‘R\ .\\‘~ 1
‘I I// \‘ _
1.0 — ‘. fi%0'0“0\\:‘ I
_ l". , ,o' 3% _
3 ”9’ \T
0.5 —- 3 ’ —
" "1
O I l I l I 1 J
O 0.25 0.50 0.75 1.00

E
O x* = 0.56, D x* = 0.62, V x* = 0.68, O x* = 0.75, I x)k = 0.82, A x* = 0.88

Figure C.4. Streamwise evolution of the total pdf for the shear layer forced at f = 4
Hz, A = 6°.

108

 

 

 

 

 

 

 

2.5 ' I l I l I I
2.0 V V
1.5 — /: A\ a
I“: _ ‘/ .l//V\"‘ d
’/V,)3 \I
1.0 V /?§ ”61“. Ci‘ J
,. ,, ' Okg
_ ‘Gt‘ _
0.5 V V
O I l I l I l I
0 0.25 0.50 0.75 1.00

E,
O x* =1.11, D x* =1.24, V )6“ =1.38, a x* =1.51, I )6“ =1.64, A )6“ =1.78

Figure C.5. Streamwise evolution of the total pdf for the shear layer forced at f = 8

 

 

 

 

 

 

 

Hz, A = 2°.
2.5 I I I I I I I
2.0 V V
- am —
g — z...\ \\ -
, «Er/EL V\ K
1.0 _ I’F ’Q”Ox\x "'1
, 9’ “
_ x x (33}: _
“1%
0.5 V .1
0 I l I l I l I
O 0.25 0.50 0.75 1.00

E
0 xi“ = 1.11, D x* = 1.24, V x* = 1.38, O x* = 1.51, I x‘ =1.64, A x' =1.78

Figure C.6. Streamwise evolution of the total pdf for the shear layer forced at f = 8
Hz, A = 6°.

APPENDIX D

Amplitude dependence of the total pdf

Amplitude dependence of the total pdf for the forced shear layers at the

downstream locations not presented in Section 3.5.2 are shown in Figures D.1 - D.15.

109

 

110

 

 

 

 

 

 

 

2.5 l l l I 1 l I
2.0 t“ _
1.5 V V
Q _ i
9.:
1.0 V V
I— \§ _
0.5 V i a
O I I I l 1 l I
0 0.25 0.50 0.75 1.00

é
o Unforced case, I3 A = 2°, V A = 4°, 0 A = 8°

Figure D.1. Amplitude dependence of the total pdf for the shear layer forced at f = 2
Hz, at x = 12.5 cm (x‘ = 0.28), compared to the unforced case.

 

 

 

 

 

 

2.5 I I I I I I I
2.0 V V
1.5 V V
g _ _
Dd .IC’J‘\
1.0 — 1’ 75' age —
_ U 1’ 4’ I“ _
l. Ing N
I g <
0.5 V %3 g’g’gx _
Z~\ ,- " - '
- t'v—oLGLQl—ar‘a” -
O l I I L I l I
O 0.25 0.50 0.75 1.00
5

o Unforced case, C! A = 2°, V A = 4°, 0 A = 8°

Figure D.2. Amplitude dependence of the total pdf for the shear layer forced at f = 2
Hz, at x = 14.0 cm (x' = 0.31), compared to the unforced case.

111

 

2.5 I I I I I I I

2.0 V V

1.5 V —

P(§)

 

 

 

 

 

 

o Unforced case, B A = 2°, V A = 4°, 0 A = 8°

Figure D.3. Amplitude dependence of the total pdf for the shear layer forced at f = 2
Hz, at x = 15.5 cm (x’ = 0.34), compared to the unforced case.

 

 

 

2.5 I I I I I I I
2.0 V _
1.5 V ,"IQ V
V‘ C’
M , - \
a: {5578“
, ' \\ I
1.0 _ 7/ \ _
I ‘\‘\
1’ \
h / Kg ‘

 

 

 

O 0.25 0.50 0.75 1.00
E
O Unforced case, B A = 2°, V A = 4°, 0 A 2 8°

Figure D.4. Amplitude dependence of the total pdf for the shear layer forced at f = 2
Hz, at x = 17.0 cm (x' = 0.37), compared to the unforced case.

112

 

 

 

 

 

 

2.5 I l l I I l I
2.0 V V
,C-

1.5 V , 1‘ a
Q l— é::g::§: \“ -I
“I g “N!

10 I— ’2’” \x \‘ ._4

’fi ®\\\i
\
I § ‘9’? \\ -
ax
0.5 V “\Q CHE/g; —
is , /
_ EFF _
O I I I l I I I
0 0.25 0.50 0.75 1.00
i

o Unforced case, B A = 2°, V A = 4°, 0 A = 8°

Figure D.5. Amplitude dependence of the total pdf for the shear layer forced at f = 2
Hz, at x = 18.5 cm (x' = 0.41), compared to the unforced case.

 

 

 

 

 

 

2.5 I I l T I 1 I
2.0 V V
1.5 V V
(g _ I
Du
1.0 — ._ —
— “ afiflffiy d
g\‘ {E
0.5 V ‘x ,5? V
II .45
_ 5.x“! ! I am? a
O I l I l I I l
O 0.25 0.50 0.75 1.00

E
o Unforced case, D A = 2°, V A = 4°, 0 A = 6°

Figure D.6. Amplitude dependence of the total pdf for the shear layer forced at f = 4
Hz, at x = 12.5 cm (x‘ = 0.56), compared to the unforced case.

 

113

 

 

 

 

 

 

 

2.5 ' I l l I I I
2.0 V V
1.5 — j —
Q» _ i _
m | -
1.0 — I Kg -
_ y f _
05 — éi' , " -
. \\‘\ A};
’ EVE-M -
O I I I I I L I
0 0.25 0.50 0.75 1.00
I:

o Unforced case, B A = 2°, V A = 4°, 0 A = 6°

Figure D.7. Amplitude dependence of the total pdf for the shear layer forced at f = 4
Hz, at x = 14.0 cm (x* = 0.62), compared to the unforced case.

 

2.5 I I I I I I I

2.0 V V

1.5 V V

P(§)

 

 

 

 

 

 

O 0.25 0.50 0.75 1.00

o Unforced case, D A = 2°, V A = 4°, 0 A = 6°

Figure D.8. Amplitude dependence of the total pdf for the shear layer forced at f = 4
Hz, at x = 15.5 cm (x* = 0.68), compared to the unforced case.

 

114

 

 

 

 

 

 

 

2.5 ' I l I I I I
2.0 V V
A 1.5 V [A V
*3 _ x V\\ _
D-I ’ V‘Q \.
10 57! \EE
. - ,; \\ _
_ g/ “\éig _
()5 — M% _
- $1”??? I
0 L I I I 4 I I
O 0.25 0.50 0.75 1.00

o Unforced case, C! A = 2°, V A = 4°, 0 A = 6°

Figure D.9. Amplitude dependence of the total pdf for the shear layer forced at f = 4
Hz, at x = 17.0 cm (x* = 0.75), compared to the unforced case.

 

2.5 I T I I I I I

1.5—II

 

 

 

 

 

 

@ _ I I
OH ‘I
\
1.0 V I. _
ERI
.. V I _
“II.
0.5 — K —
— \ 3246—? —
O I I I J I I I
0 0.25 0.50 0.75 1.00

é
o Unforced case, C! A = 2°, V A = 4°, 0 A = 6°

Figure D. 10. Amplitude dependence of the total pdf for the shear layer forced at f = 4
Hz, at x = 18.5 cm (x* = 0.82), compared to the unforced case.

 

115
2.5 . I I I I I

 

 

 

 

 

 

I ' '
l
2.0 — I _
_ I _
I
I
1.5 V I V
(Cr I
E ' I I
1.0 — ‘ —
I :gfi—V—Efi
— [Q I‘ fil/ \& a _
_ II‘\I Q: _
0.5 \I ,9!"
$4.55“
- Ragga—W -
0 I I I | I | 41
O 0 25 0.50 0.75 1 00
Q

o Unforced case, B A = 2°, V A = 4°, 0 A = 6°

Figure D.11. Amplitude dependence of the total pdf for the shear layer forced at f = 8
Hz, at x = 12.5 cm (x‘ = 1.11), compared to the unforced case.

2.5 I I I I

 

I I

2.0 V

Ni)

 

 

VL} \l'

0.5 V

 

 

 

 

0 0.25 0.50 0.75 1.00

o Unforced case, D A = 2°, V A = 4°, 0 A = 6°

Figure D.12. Amplitude dependence of the total pdf for the shear layer forced at f = 8
Hz, at x = 14.0 cm (x' = 1.24), compared to the unforced case.

116

 

 

 

 

 

 

2.5 ' I I I I I l
2.0 V V
1.5 V V
e _ ”x —
D... /
gv’a §\ ‘\
10 V [pi/I Q WV 0 —
- 59’: O ISM? _
11?? ‘W
0.5 V IE Egg/157 o _
_ ‘3g-gg—géggggffi-Vg'? _
0 I | L l I | I
0 0.25 0.50 0.75 1.00
E

O Unforced case, B A = 2°, V A = 4°, 0 A = 6°

Figure D.13. Amplitude dependence of the total pdf for the shear layer forced at f = 8
Hz, at x = 15.5 cm (x‘ = 1.38), compared to the unforced case.

 

 

 

 

 

 

2.5 7 VI I I I I I
2.0 V V
_ f\ J
A 15 V iv ‘R
g V I / ‘0 NR \\ V
10 — EI , WK" _
II fix”; Q‘I ISL \
- 9‘1. % off] I
I,\\\ / ‘o
0.5 — “' jg ,: Va a
‘ — zEl'
Eatmifi -
0 I I I I I I I
0 0.25 0.50 0.75 1.00
E

o Unforced case, D A = 2°, V A = 4°, 0 A = 6°

Figure D.14. Amplitude dependence of the total pdf for the shear layer forced at f = 8
Hz, at x = 17.0 cm (x* = 1.51), compared to the unforced case.

 

 

 

117

 

 

 

 

 

 

2.5 I. ' l I I I I I
_ I _
2.0 V _
_ li/‘ _
\ I
1.5 V WV \ V
A / ‘ \ \\
2% ~ {° EI‘ \‘I ‘
, , o .
/ x
1.0 — 17% \Q \m‘ '—
V \\ Ayflf \‘Q I V
X /.”EI/
05 .. ‘\\\I /.§ ’1“ E _
. \I - — _ _. _, $33E/g/B/
_ E§§g§&@ _
O I J_ I I I I J
O 0.25 0.50 0.75 1.00
E

O Unforced case, C! A = 2°, V A = 4°, 0 A = 6°

Figure D.15. Amplitude dependence of the total pdf for the shear layer forced at f = 8
Hz, at x = 18.5 cm (x’ = 1.64), compared to the unforced case.

 

 

REFERENCES

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